Number of stress cycles =============================== Reminders: spectral moments and the irregularity factor ----------------------------------------------------- The following quantity [:ref:`R7.10.01 `] is called the spectral moment of order :math:`i`: :math:`{\lambda }_{i}\text{=}\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{\mid \omega \mid }^{i}{G}_{\text{SS}}(\omega )d\omega` where :math:`\omega` is the pulsation and :math:`{G}_{\text{SS}}` is the power spectral density or DSP. In particular, we have: :math:`{\lambda }_{0}\text{=}{s}_{S}^{2}{\lambda }_{2}={s}_{{S}^{\text{'}}}^{2}{\lambda }_{4}={s}_{{S}^{\text{'}\text{'}}}^{2}` which are the standard deviations of :math:`S` and its first derivatives. The **irregularity factor** reflects the frequency pattern of the signal. Between 0 and 1, it tends to 1 when the process is narrow band, on the other hand it tends to 0 for a broadband process. His expression is: :math:`I\text{=}\frac{{s}_{{S}^{\text{'}}}^{2}}{{s}_{{S}^{}}{s}_{{S}^{\text{''}}}}\text{=}\sqrt{\frac{{\lambda }_{2}^{2}}{{\lambda }_{0}{\lambda }_{4}}}` We recall these definitions because the evolution of the signal is assimilated to a random process characterized by its statistical parameters (spectral moments of order 0, 2 and 4). For the method for counting stress peaks, the random signal is entirely characterized by the three spectral moments of order 0, 2 and 4. In the case of the method for counting level overruns, spectral moments of order 0 and 2 are sufficient to characterize the random signal. In practice, these values are determined by the command POST_DYNA_ALEA [:external:ref:`U4.76.02 `] which performs statistical processing on a random loading. The definition of the various parameters is given in the document [:ref:`R7.10.01 `]. The random domain fatigue calculation operator POST_FATI_ALEA [:external:ref:`U4.67.05 `] uses the values of the three spectral moments calculated by POST_DYNA_ALEA and calculates the mean damage and the standard deviation of the damage by the methods described in this document. Method for counting stress peaks ------------------------------------------- The principle of this method consists in counting the local maximums (in absolute value) located on either side of the mean value of the constraints. Since the signal is Gaussian stationary and centered with respect to its mean value, the distribution of the peaks is symmetric with respect to this mean. We are therefore interested in the **distribution of positive peaks**. In the general case, the distribution of peaks with a positive amplitude :math:`S` is written in the form: :math:`{P}_{\text{pic}}^{\text{+}}\left(S\right)\text{=}\frac{2}{\sqrt{2\pi }{s}_{S}\left(1\text{+}I\right)}\left[\sqrt{1\text{-}{I}^{2}}{e}^{\text{}\frac{{S}^{2}}{{\mathrm{2s}}_{S}^{2}\left(1\text{-}{I}^{2}\right)}}\text{+}\frac{\text{IS}}{{s}_{S}}{e}^{\text{}\frac{{S}^{2}}{{\mathrm{2s}}_{S}^{2}}}\underset{\text{-}\infty }{\overset{\alpha }{\int }}{e}^{\text{}\frac{-{t}^{2}}{2}}\text{dt}\right]` :math:`\text{avec}\{\begin{array}{c}I\text{=}\frac{{\sigma }_{{S}^{\text{'}}}^{2}}{{\sigma }_{S}{\sigma }_{{S}^{\text{'}\text{'}}}}\\ \alpha \text{=}\frac{S}{{\sigma }_{S}}\frac{I}{\sqrt{1\text{-}{I}^{2}}}\end{array}` This distribution of positive peaks is simplified in the case of signals for which the irregularity factor is equal to :math:`I\mathrm{=}0` or :math:`I\mathrm{=}1`. : * Broadband signal: Gauss law or normal law :math:`(I\text{=}0)` :math:`{P}_{\text{pic}}^{\text{+}}\left(S\right)\text{=}\frac{2}{\sqrt{2\pi {\sigma }_{{S}^{2}}}}{e}^{\text{-}\frac{{S}^{2}}{2{\sigma }_{{S}^{2}}}}` * Narrow band signal: Rayleigh law :math:`(I\text{=}1)` :math:`{P}_{\text{pic}}^{\text{+}}(S)\text{=}\frac{S}{{\sigma }_{{S}^{2}}}{e}^{\text{-}\frac{{S}^{2}}{2{\sigma }_{{S}^{2}}}}` The method for counting stress peaks associates each positive amplitude peak :math:`S` with a cycle of amplitude :math:`\Delta S\text{=}2S` (we therefore directly have :math:`S\text{=}\text{Salt}`). The number of positive amplitude peaks is written as: :math:`{n}_{\text{pic}}^{\text{+}}(S)\text{=}{P}_{\text{pic}}^{\text{+}}(S)\times {N}_{\text{pic}}^{\text{+}}` where :math:`{N}_{\text{pic}}^{\text{+}}\text{=}\frac{1}{4}(1\text{+}I)\times \frac{1}{\pi }\frac{\sigma {S}^{\text{'}\text{'}}}{{\sigma }_{{S}^{\text{'}}}}` = average number of positive peaks per unit of time. Hence the :math:`N` number of cycles to take into account is: :math:`N(S)\text{=}\frac{1\text{+}I}{4\pi }\frac{{\sigma }_{{S}^{\text{'}\text{'}}}}{{\sigma }_{{S}^{\text{'}}}}{P}_{\text{pic}}^{\text{+}}(S)` **Note:** *Note that the expression for the number* :math:`N` *of cycles to be taken into account only depends on* :math:`{s}_{S}` *(for the calculation of the irregularity factor* :math:`I` *),* :math:`{\sigma }_{{S}^{\text{'}}}` *and* :math:`{\sigma }_{{S}^{\text{'}\text{'}}}` *.* Method for counting level breaches ---------------------------------------------- This method requires the division of stress variations into amplitude classes. The number of cycles :math:`N(S)` is obtained from the difference in the numbers of level overruns with a positive slope between two successive classes, starting from the maximum amplitude class. For a centered Gaussian process the expression for :math:`N(S)` is: :math:`N(S)\text{=}\frac{1}{\pi }\frac{{\sigma }_{{S}^{\text{'}}}}{{\sigma }_{S}^{}}{e}^{\text{-}\frac{{S}^{2}}{2{\sigma }_{S}^{2}}}` **Notes:** *Expressing the number* :math:`N` *of cycles to be taken into account requires only the knowledge of* :math:`{\sigma }_{S}` *and* :math:`{\sigma }_{{S}^{\text{'}}}` *(independence with* :math:`{\sigma }_{{S}^{\text{'}\text{'}}}` *) .* *The irregularity coefficient is not involved in this method* :math:`I` *.*