PSI - Issue 54

Arvid Trapp et al. / Procedia Structural Integrity 54 (2024) 521–535 Arvid Trapp / Structural Integrity Procedia 00 (2023) 000–000

525

5

This equation accumulates all stress cycles in the form of partial damages n j s k j , from which an equivalent response amplitude s eq is derived. The equivalent response amplitude causes the same damage considering n eq cycles under linear damage accumulation using the elementary Palmgren-Miner rule. This can be interpreted as converting the load spectrum of variable amplitudes into a single-level spectrum of n eq cycles for an S-N curve exponent k . The equivalent response amplitude provides a concise way to compare di ff erent approaches or methods for a fatigue assessment. This concept is commonly referred to as pseudo damage. Fundamental to a statistical fatigue assessment is the power spectral density (PSD). The PSD G xx ( f ) is the frequency-domain decomposition of the variance σ 2 . Respectively, formalized by central moments µ n , the second order central moment µ 2 = σ 2 =  ∞ 0 G xx ( f )d f = λ 0 and consequently fully characterizes stationary Gaussian loading with zero mean µ = 0 by its Gaussian probability density function (PDF)

   −

2 σ 2   

( x − µ ) 2

1 √ 2 πσ 2

p g ( x ) =

(2)

e

The frequency-domain characterization via PSD allows for an e ffi cient statistical characterization of random vibration loading as a continuous function. As such, it requires severely less data in capturing random loading than a time domain realization x ( t ). This pays of, when calculating response PSDs via linear systems theory for the structure under consideration: G yy ( f ) = | H xy ( f ) | 2 G xx ( f ). The second and even more relevant time saver are spectral damage estimators such as the analytically narrowband-, and also empirical formulations such as Dirlik (DK) and Tovo Benasciutti (Dirlik (1985); Benasciutti (2004)). These are based on the information that is accessible via spectral moments (Lutes and Sarkani (2004); Slavicˇ et al. (2020)) λ n =  ∞ 0 (2 π f ) n G xx ( f )d f (3) Spectral moments λ n relate to essential time-domain metrics, such as the upward zero-crossing- ν + 0 = 1 / 2 π √ λ 2 /λ 0 and peak-rate ν p = 1 / 2 π √ λ 4 /λ 2 , but also to the derivatives of the process (Lutes and Sarkani (2004)), e.g. λ 2 cor responds the area below G ˙ x ˙ x ( f ). Exemplary, the equivalent response amplitude can then be calculated via the nar rowband approach s (NB) eq = ν 0 T ( √ 2 λ 0 ) k Γ (1 + k 2 ) while methods such as Dirlik s (DK) eq , which have been designed for wideband stationary Gaussian processes, require the calculation of a few additional parameters (Dirlik (1985); Dirlik and Benasciutti (2021)). For stationary Gaussian loading s (RFC) eq and s (DK) eq generally deviates insignificantly. But as soon as stress states are non-Gaussian, s (DK) eq starts to deviate, in most cases non-conservatively. In the consideration of characteristics that make loading non-Gaussian — quite generally meaning a PDF that deviates from Eq. (2) — non-stationarity is critical as linear structures whose modes are excited with varying intensity will reproduce this non stationarity clearly in its responses (Palmieri et al. (2017)). A concise and very popular descriptor for non-Gaussianity is the kurtosis β , which is the standardized fourth-order moment

f 1  f 2

β ≈ 

M xx ( f 1 , f 2 ) d f 1 d f 2 (  f G xx ( f ) d f ) 2

µ 4 ( µ 2 )

2 ;

(4)

β =

As highlighted in the last passage, critical for fatigue damage is if relevant kurtosis — i.e. non-Gaussianity — transfers from loading β x into stress states kurtosis β y . However, this information has not been accessible when conduction a PSD-based statistical fatigue assessment. Therefore, this section continues with the non-stationarity matrix (NSM) M xx ( f 1 , f 2 ) which provides a spectral representation for µ 4 under the assumption of a slowly-varying non-stationary process. In this case µ 4 ≈  f 1  f 2 M xx ( f 1 , f 2 )d f 1 d f 2 and as such the NSM extends the spectral second-order average