20191201_00414173.pdf - Fatigue analysis of non-Gaussian...

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Fatigue analysis of non-Gaussian random loadings 153 mean upcrossings, peaks or valleys at the same instants of time, respectively; then, both processes will possess the same irregularity factor IF . Referring in particular to the correspondence between peaks and valleys, if ) ( t X has a peak at level p x at time i t , ) ( t Z will have a corresponding peak at level ( ) ) ( p p i t x G z = at the same instant of time; similarly, a trough v x is trans- formed in ( ) ) ( v v i t x G z = . Furthermore, imposing the transformation as being monotonically non- decreasing allow us also to preserve the relative position amongst peaks and val- leys; for example, if ) ( ) ( 2 p 1 p t x t x > , then it will be ) ( ) ( 2 p 1 p t z t z > , and the same for valleys. Consequently, since in a random process the rainflow counting extracts cycles by pairing peaks and valleys based on their relative positions and time sequence (so forming complete cycles or half-cycles), rainflow counted cy- cles in the ) ( t X and transformed ) ( t Z process will be counted at the same in- stants of time, and they will be formed by coupling pairs of corresponding peaks and valleys. In other words, the transformation, without changing the number and the sequence of rainflow counted cycles, modifies instead their amplitudes and mean values by increasing or decreasing their maximum and minimum values. This fact then enables us to establish a fundamental relation between a rainflow cycle counted in the ) ( t X process and its correspondent cycle in the transformed ) ( t Z process: given a cycle in process ) ( t X having peak ) ( p i t x and valley ) ( v j t x , we can associate to it a corresponding cycle in process ) ( t Z , having peak ) ( p i t z and valley ) ( v j t
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