Moments of the Power Spectral Density

From the characteristics of the PSD, nth moments M, can be obtained. The relevant spectral moments Mj are easily computed from a one sided power spectral density, W(f), Eq. (8.78) and Fig. 8.42, using the following expression 

Summation in Eq. (8.80) can be used in case where the PSD is not a continuous function but concentrates in more or less frequencies, as shown in Fig. 8.44. The determination of the moments Mj is fundamental in the fatigue analysis since it makes it possible to calculate an empirical closed-form expression for the pdf probability density function of a random spectrum variable X(t) [35], the number of zero up crossing [0] and peak [P] per second in the sample period T and the irregularity factor y given by Eq. (8.61). In theory, all the possible moments could be calculated, however, in practice, M , A/i, M2 and M4 are sufficient to calculate all of the information for the fatigue analysis. This information is  

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