The spectral moments (\lambda_n) are central to fatigue metrics:
[ \lambda_n = \int_0^\infty f^n , G_\sigma\sigma(f) , df, \quad n = 0,1,2,4 ]
[ E[D] \textWL = \rho(b,\gamma) \cdot E[D] \textNarrowband ] [ \rho(b,\gamma) = a(b) + 1 - a(b) ^c(b) ] [ a(b) = 0.926 - 0.033b, \quad c(b) = 1.587b - 2.323 ] Widely used in commercial software (e.g., nCode, FEMFAT). Empirically fits the rainflow cycle amplitude distribution as a sum of one exponential and two Rayleigh distributions: vibration fatigue by spectral methods pdf
| Method | Damage per sec | Lifetime (hours) | |---------------|----------------|------------------| | Time-domain RF| (3.2 \times 10^-8) | 8680 | | Narrowband | (7.1 \times 10^-8) | 3910 (underest.)| | Dirlik | (3.5 \times 10^-8) | 7930 (error 8.6%)|
(\lambda_0, \lambda_1, \lambda_2, \lambda_4) via numerical integration over frequency range. The spectral moments (\lambda_n) are central to fatigue
where (\Gamma) is the gamma function. This is for broadband signals. 4. Broadband Spectral Fatigue Criteria To address broadband processes, several frequency-domain methods have been developed: 4.1 Wirsching–Light (WL) Method Applies a correction factor (\rho(b,\gamma)) to the narrowband damage:
[ E[\sigma^2] = \int_0^\infty G_\sigma\sigma(f) , df ] This is for broadband signals
[ E[D] \textDK = f_p , C^-1 \int 0^\infty S^b , p_\textDK(S) , dS ] | Method | Accuracy (broadband) | Computational cost | Best suited for | |----------------|----------------------|--------------------|---------------------------| | Narrowband | Poor (conservative) | Very low | Nearly sinusoidal stress | | Wirsching-Light| Moderate | Low | Offshore/wind structures | | Dirlik | High (error <10%) | Moderate | General random vibration | | Zhao-Baker | High | Moderate | Bimodal spectra | 5. Practical Procedure for Spectral Fatigue Analysis Step 1: Obtain stress PSD From finite element analysis (modal or direct frequency response) or experimental measurements (strain gauge + FFT).