Assessment of non-stationary random vibration loading (#27)
A. Trapp1, P. Wolfsteiner1
1 University of Applied Sciences, Department of Mechanical, Automotive and Aeronautical Engineering, Munich, Bavaria, Germany
Designing mechanical structures exposed to random vibration loading compromises two central challenges. The first is to abstract representative load assumptions for the in-service use. The second is to efficiently process these load assumptions in a fatigue assessment. Both are primarily a matter of statistical load description.
Defining random loading via the power spectral density (PSD) combines an effective statistical description with the ability to efficiently perform structural response analyses of linear structures and subsequent fatigue analyses using frequency-domain methods (e.g. Dirlik). The PSD covers a full stochastic description for stationary Gaussian random processes. Measured in-service vibration loading, such as road, railway, and aerospace vehicle excitation, usually differ significantly from this assumption. In particular, the constant change of operating, environmental and excitation source conditions causes complex loading, consisting of superimposed vibration states with varying intensity. Employing the PSD for such non-stationary loading results in the averaged power spectral density. A replication of the loading solely based on the averaged PSD generally has lower amplitudes than the referencing measured load series. Thus processing the PSD for non-stationary loading leads to non-conservative deviations for the lifetime estimation.
Researchers have addressed this problem by extending the description of the PSD with measures that differ significantly in complexity. On the one hand higher-order statistical moments, foremost the kurtosis, specifying the probability density function, on the other hand the short-time Fourier transform, resolving the frequency content in time. While the former does not relate to frequency and lacks to relate to structural response behavior, the latter produce too much data to be processed in reasonable time.
We propose a measure that finds a middle way. This measure is a spectral correlation matrix, which contains the time-averaged interaction between frequency contributions. This measure can be related to the trispectrum, the spectral decomposition of kurtosis, and thus directly relates to probabilistic measures.
Initially this measure allows identifying whether a load is stationary or non-stationary. If so, superimposed loading states can be differentiated to some extent and their variation in time can be quantified. Our contribution will present (i) the derivation of the spectral correlation matrix from sampled load series (ii) how to test for stationarity and superimposed loading states (iii) the link between non-stationarity and fatigue damage.
Keywords: non-stationary loading, random vibration loading, spectral correlation
A review on the methods for modelling loading spectra and their scatter (#34)
M. Nagode1, J. Klemenc1
1 University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia
State-of-the-art product development demands also a thorough insight into the usage, environmental and maintenance conditions to which the product is going to be exposed during its service life. A product’s service life with a certain reliability can be estimated in early development stages presuming that loading, materials and technologies are known. The paper focuses on structural loads.
Structural loads can be assessed by in-field measurements, measurements carried out on a test track or simulations. In all of the cases, the measurements or simulations do not take into account the complete service live nor exactly reflect the actual service loading. This means that the structural loads should be somewhat modelled and extrapolated for the purposes of fatigue-life or reliability prediction.
Supposing that the load represents a stochastic process and knowing that the service life of dynamically loaded products largely depends on fatigue, the loading histories are commonly transferred into loading spectra. For this purpose the rainflow counting has become standard. The loading histories are thus transferred into either one-dimensional histograms of load cycle amplitudes or two-dimensional histograms of load cycle amplitudes and load cycle means. The latter can also be transferred into the so called from-to rainflow matrix.
Once the empirical densities are known, the next commonly employed step is a probability density estimation. Probability densities are required for proper fatigue-life and reliability predictions. Non-parametric and parametric probability density estimation techniques are available and have been increasingly used in various scientific fields.
The paper gives a thorough review of the available non-parametric and parametric probability, density estimation techniques and the corresponding publically available software. In the paper several typical loading histories attained by measurements on different vehicle types are going to be transformed into univariate and bivariate histograms by using the rainflow counting and then modelled by a number of available non-parametric and parametric probability density estimation approaches built into the publically available software. Here different parametric families like normal, lognormal, Weibull and gamma are considered. Different approaches are going to be evaluated regarding precision, accuracy and computational time. After the probability densities of the rainflow load cycles are estimated, the corresponding loading spectra will be extrapolated and their expected scatter will be assessed for the selected load cases.
Keywords: loading spectrum, probability density function, mixed distributions, normal pdf, Weibull pdf
Impact of the kurtosis parameter of the load on the fatigue life of a structure (#66)
A. J. Niesłony1, M. Böhm1, R. Owsiński1
1 Opole University of Technology, Department of Mechanics and Machine Design, Opole, Poland
Component tests on electromagnetic shakers are often used in industry . Such tests show the resistance of components to shocks and vibrations and they are used to determine fatigue life. Duration of tests is important, which significant affects testing costs. For this reason, specially prepared load histories are used to cause fatigue damage in the tested structure in a sufficiently short time. One of the methods is to change the distribution of the loading from Gaussian to non-Gaussian using the kurtosis parameter . Properly prepared load course has greater damage potential in time than the Gaussian course having the same standard deviation. In this article, the authors will show that non-Gaussian loads also affect the loadings measured on the tested elements. This is important because the deviation from the normal distribution should be corrected during the calculation. In particular, it is important for the spectral methods for fatigue life calculation, where the Gaussian load distribution is assumed [3,4].
 A. Steinwolf, "Vibration testing of vehicle components by random excitations with increased kurtosis", International Journal of Vehicle Noise and Vibration, vol. 11, no. 1, p. 39, 2015.
 F. Kihm, N. S. Ferguson, and J. Antoni, "Fatigue Life from Kurtosis Controlled Excitations", Procedia Engineering, vol. 133, no. Supplement C, pp. 698–713, Jan. 2015.
 A. Niesłony, M. Böhm, T. Łagoda, and F. Cianetti, "The use of spectral method for fatigue life assessment for non-Gaussian random loads", Acta Mechanica et Automatica, vol. 10, no. 2, pp. 100–103, 2016.
 C. Braccesi, F. Cianetti, G. Lori, and D. Pioli, "The frequency domain approach in virtual fatigue estimation of non-linear systems: The problem of non-Gaussian states of stress", International Journal of Fatigue, vol. 31, no. 4, pp. 766–775, Apr. 2009.
Keywords: kurtosis, spectral method, fatigue testing on shaker
Extrapolation of load spectra using Kernel Density Estimators (#108)
V. Schröder1, C. Müller2, A. Esderts1
1 Clausthal University of Technology, Institute for Plant Engineering and Fatigue Analysis, Clausthal-Zellerfeld, Lower Saxony, Germany
With the aim of safe load assumptions, measured load spectra need to be shifted to – usually larger – frequencies that represent the intended life expectancy. Therefore, these load spectra are usually right shifted to a specific lifecycle range. The overall frequency may be accurately extrapolated by this. However, this results in an undefined gap between the ordinate axis and the maximum of the right shifted load spectra. An approach for closing this gap and examining the right shifted load spectra shall be tried with the following steps:
Following these steps a known population is generated that can be used to rate different influence factors, e.g. different kernel functions and bandwidth, within the described extrapolation method. By selecting the bandwidth, the sensitivity of the kernel density estimation can be adjusted; the larger the bandwidth, the smoother the course of the probability density function.
With the best kernel function and the optimum of the parameter bandwidth, the procedure is compared to real measurement data. Finally, a recommendation is given to the user for appropriate parameters for kernel density estimation and extrapolation.
Keywords: Extrapolation, load spectra, Kernel Density Estimator