PSI - Issue 75

Sébastien Boudevin et al. / Procedia Structural Integrity 75 (2025) 72–84 Author name / Structural Integrity Procedia (2025)

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One way of characterizing any random variable is the probability density function or PDF, using a distribution such as normal, log-normal, discrete, Weibull, or Rayleigh. The PDF can be applied to characterize the time series loading signal, with the mean value and standard deviation in the case of the normal distribution. Distribution fitting methods may include rank regression or maximum likelihood estimation (MLE). The PDF can also be calculated on each bin of a histogram, instead of considering all the data points in the time signals. The usage of a data lake provides a convenient way to search the database, retrieve events and such calculated histograms associated with parent events, and at the end to get mission profile or histogram PDF. As described in NF X 50-144-3 (2021), for each value of the histogram, the damage at each frequency for example, the PDF parameters can be adjusted, and from a percentile value p=1-a, we can derive a design-focused value of FDSa. The way to

calculate these FDSa is described in Table 1. Table 1. Computation of severized histograms. Weibull

Normal

Log-Normal

Rayleigh

( − ) −1 −( − ) √12 −12( − ) 2 ′ = ln ( ) ′√ 1 2 −12( ′− ′ ′) 2 ( − 2 ) −12( − ) 2

Probability Density Function Cumulative Density Function

f(t)

F(t) 1− −( − ) ∫ √12 −12( − ) 2 −∞ −( − ) ∫ √12 −12( − ) 2 ∞ t(R=  ) + . (− l (1 − ) ) 1/ + . −1 ( ) R(t)

∫ ′√12 −12( − ′ ′ ) ′ −∞ ∫ ′√12 −12( − ′ ′ ) ∞ ′

2 2

1− −12( − ) 2 −12( − ) 2

Reliability

FDS  ( ′+ ′. −1 ( )) + √−2ln (1− ) The FDSa is designed to characterize the non-stationarity nature of a signal, using the MBD method, and can also be applied to several events to quantify load uncertainties. If several tests are merged as part of the duty cycle, a time/distance weighted average operation must be performed, before adjusting the PDF parameters for each histogram class. From this fatigue damage spectrum, design-specific to the target percentile value, a synthetic random PSD can be built. This vibration profile can then be used for analytical life prediction and physical durability validation testing. The example below shows the FDSa based on several events in Fig. 1.

Fig. 1. FDSa (in bold red) obtained from the FDS of 7 events, at 90% percentile value