By Adam Nieslony, Ewald Macha
This monograph comprises theoretical foundations of the spectral technique for fatigue existence decision the place the authors speak about a rule of description of random loading states with the matrix of energy spectral density services of the stress/strain tensor parts. a few selected standards of multiaxial fatigue failure being linear mixtures of rigidity or pressure parts at the serious aircraft are analyzed.
The formulation proposed during this e-book allows to figure out energy spectral density of the an identical heritage without delay from the parts of the facility spectral density matrix of the multidimensional stochastic technique. It offers the assumptions and the technique of choice of easy relationships of the spectral technique. The authors determine equations picking the fatigue lifestyles in response to the spectral process utilizing a variety of linear hypotheses of fatigue harm accumulation. The set of rules of fatigue lifestyles includes 5 blocks: 1 – decision of loading, 2 – selection of the serious aircraft place for the assumed multiaxial fatigue failure criterion, three – choice of energy spectral density of the similar tension or pressure, four – decision of statistical parameters of the an identical parameter chargeable for fatigue harm, and five – fatigue existence calculation in keeping with an appropriate speculation of wear accumulation.
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Extra info for Spectral method in multiaxial random fatigue
14) ⎦. . G61 (f ) · · · G66 (f ) The functions Gkl (f ) are deﬁned for frequency f ≥ 0 and are equal to the double value of two-sided power spectral density Skl (f ) Gkl (f ) = 2Skl (f ) 0 for for 0 ≤ f < ∞, f < 0, (k, l = 1, . . 15) where: Gkk (f ), Skk (f ) – autospectral density functions of component processes Xk (t), Gkl (f ), Skl (f ) – cross-spectral density functions between component processes Xk (t) and Xl (t). 16) where: Re[Gkl (f )] – coincident spectral density function, a real part of Gkl (f ), Im[Gkl (f )] – quadrature spectral density function, an imaginary part of Gkl (f ), √ i = −1 – imaginary unit.
Damage accumulation is often applied only for amplitudes above a speciﬁed value in accordance with the conviction that amplitudes below some limit do not aﬀect material damage considerably. The limit is associated with fatigue limit σaf , and accumulation of damage is conducted in accordance with the modiﬁed Palmgren-Miner hypothesis. The hypothesis is illustrated in Fig. 3. σa σa = A Nf 1 m σa max σaf p(σa ) aP M σaf Nf p(σa ) Fig. 3. Damage accumulation in accordance with modiﬁed Palmgren-Miner hypothesis.
The probability of both events is following 48 3 Theoretical Fundamentals x(t) upward sloping function x(t) downward sloping function x(t) a 0 τ τ + dt t Fig. 2. A section of random history x(t) ∞ a p(x, x)dxd ˙ x˙ . 38) 0 a−xdt ˙ Under the above assumptions a formula for mean number of level x = a crossings Na+ in time unit is derived, ﬁrst quoted by Rice in 1945  ∞ Na+ = xp(a, ˙ x)d ˙ x˙ . 39) 0 Under the assumption that loading is a random history x(t) with normal probability distribution of instantaneous values, the distribution of the mean number of a level crossing in second could be deﬁned.