Vibration fatigue of a vibration fatigue analysis is the modal analysis, that exposes the natural modes and frequencies of the vibrating structure and enables accurate prediction of the local stress responses for the given excitation. Only then, when the stress responses are known, can vibration fatigue be successfully characterized. The more classical approach of fatigue evaluation consists of cycle counting, using the rainflow algorithm and summation by means of the Palmgren-Miner linear damage hypothesis, that appropriately sums the damages of respective cycles. When the time history is not known, because the load is random ("e.g." a car on a rough road or
Parity of a permutation of formula_1 into disjoint cycles, which can be composed in any order because they commute. A cycle formula_49 involving formula_50 points can always be obtained by composing formula_51 transpositions (2-cycles): so call formula_51 the "size" of the cycle, and observe that, under this definition, transpositions are cycles of size 1. From a decomposition into formula_54 disjoint cycles we can obtain a decomposition of formula_1 into formula_56 transpositions, where formula_57 is the size of the th cycle. The number formula_58 is called the discriminant of formula_1, and can also be computed as if we take care to include the fixed points