PSI - Issue 19

532 Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 5 system. The input excitation is represented as the power spectrum density, G xx ( ) , and the output response PSD by G yy ( ) , the relationship between G xx ( ) and G yy ( ) is defined by Eq. (3) considering that both input and response PSDs time depended [20,21]. The schematic representation of time varying response PSD, G yy ( ) is illustrated in Fig. 3. G yy ( ) = 2 ( ) (3) Similar to the input PSD, the response PSD can also be decomposed into u number of discrete PSD positions as shown in Fig. 3 [21]. The response PSD can be formulated in Eq. (4). G yy i ( ) = 1 + (2 ) 2 (1 − 2 ) 2 + (2 ) 2 ( ) (4) Where G yy i ( ) is the i-th number of the response PSD and i = 1… u (a number of PSD positions). The input PSD is comprised of the wideband (in a range of 5 Hz to 2000 Hz) and three narrowbands that have been swept within the wideband range as given in Table 1. Each response PSD position ( ) can be divided into a finite number of narrow frequency bands of the PSD segment [21]. Each segment represents a narrowband PSD G j ( ) with the central frequency located. The PSD division thus transforms the given discrete PSD, G(f) into a finite set of narrowband PSD segments , G j ( ) , where j = 1, 2, … v [24]. To simplify the computational modeling formulation and reduce solution time, each PSD position is further divided into a narrow frequency band of 1 Hz. Therefore, each of the u PSD decompositions described by Eq. (4) can be further split into the v number of PSD narrowband segments as illustrated in Fig. 3.

Fig. 3. Division of a given response PSD into finite number of narrowbands.

2.3. Rayleigh Probability Distribution of Stresses If each discrete response PSD function is split into v number of narrowband PSD segments, and each narrowband PSD segment can be associated with a different Rayleigh distribution of stress cycles [20,21]. Each j-th narrowband segment can be defined by a central frequency fj (where j is the number of the narrowband PSD segment), results in a different cycle as a function of fj. The Rayleigh distribution representing the narrowband PSD segment can thus be used to determine stress amplitudes and cycles. Each of the v narrowband PSD segments can thus be characterized by a distinct Rayleigh distribution which defines both the stress amplitudes and a number of stress cycles as shown in Fig. 4. The applied cyclic stresses associated with each Rayleigh distribution can be further split into k distinct stress amplitudes (e.g. five different stress amplitude and corresponding cycles as shown in Fig.4). S k represent the expectation of the applied stress amplitude for each stress region.