PSI - Issue 75

Mohamed El Yazrhi et al. / Procedia Structural Integrity 75 (2025) 262–275 Mohamed El Yazrhi , Jean-Yves Disson / Structural Integrity Procedia (2025)

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Nomenclature C, b

Parameters of ‘Basquin’ law: N.  b = C

CLT

Central Limit Theorem

K

Constant coefficient between stress and relative displacement of the SDOF oscillator model

ERS FDS

Extreme Response Spectrum Fatigue Damage Spectrum

FDS mon FDS pre

Fatigue Damage Spectrum from monitoring Fatigue Damage Spectrum predicted

FDS threshold

Fatigue Damage Spectrum not to exceed, as the FDS used for the qualification test

GMM

Gaussian Mixture Model Extrapolation factor over time Density probability function Single Degree Of Freedom

M p()

SDOF

XFS XRS

Fatigue Damage Spectra at a given exceeding risk Extreme Response Spectra at a given exceeding risk Up-crossing risk retained for equipment considered

 ẍ

(t)

Measured acceleration environment introduces at the base of the equipment and the oscillator model

x(t)

Relative displacement of the SDOF oscillator model

2. Theoretical Background We model the measured acceleration signal ( ) through a bank of linear SDOF oscillators, each with natural frequency ω and damping ratio ζ . The relative displacement x(t) of the ith oscillator satisfies ̈ +2ζω ̇ +ω 2 = ( ) or, in the frequency domain, X i (ω)=H i (ω)A(ω) with H i (ω)= -ω 1 2 +2jζ i ω+ω i 2 . Mapping displacement to material stress uses a proportionality (e.g. from the stiffness of the oscillator) so that σ i (t)=K x i (t). A rainflow counting of each stress cycle is performed, which is a time domain process, providing cycles for each class (i), the damage Di is calculated: D i = n i N i = C.n i .K b .x i b C and b are the parameters of the damage law, respectively the constant and the exponent of the law. Then, the damage Dc is cumulated according the linear Miner law: D c = ∑D i FDS j = D c j Complementarily, the Extreme Response Spectrum (ERS) is defined by the maximum displacement of each oscillator whose natural frequency is plotted in abscissa. =max (| ( )|)