PSI - Issue 37

Arvid Trapp et al. / Procedia Structural Integrity 37 (2022) 622–631 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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7

(b)

(a)

4 ( ) [ 2 4 ]

4 ( ) [ 2 4 ]

Figure 4: Cumulant trispectra of (a) HP load and (b) CN load

3.2. Linear system theory applied to higher-order spectra To determine the structural response ( ) ⊶ ( ) of an arbitrary mechanical system, the excitational load ( ) ⊶ ( ) is correlated with the corresponding transfer function ( ) : ( ) = ( ) ( ) . Carrying linear systems theory to higher-order spectra can be interpreted as defining an -th order transfer spectrum ( ) ( 1 , … , −1 ) , following the same computational procedure as the respective moment spectrum (Eq. (9). In contrast it is based on the transfer function ( ) . Resulting response spectra are computed (Eq. (10) by the multiplication of input spectra with transfer spectra of same order , which is tremendously more efficient than estimating the response spectrum from ( ) . The formulas relating excitation and response were initially provided by Brillinger et al. in (Brillinger and Rosenblatt, 1967): ( ) ( 1 , … , −1 ) = ( 1 ) ( 2 ) … ( −1 ) ∗ ( 1 + 2 + ⋯+ −1 ) (9) ( ) ( 1 , … , −1 ) = ( ) ( 1 , … , −1 ) ( ) ( 1 , … , −1 ) ( ) ( 1 , … , −1 ) = ( ) ( 1 , … , −1 ) ( ) ( 1 , … , −1 ) (10) Applied to fourth-order spectra, the transfer spectrum, and the cumulant and trispectrum are defined by: 4 ( ) ( 1 , 2 , 3 ) = ( 1 ) ( 2 ) ( 3 ) ∗ ( 1 + 2 + 3 ) 4 ( ) ( 1 , 2 , 3 ) = 4 ( ) ( 1 , 2 , 3 ) 4 ( ) ( 1 , 2 , 3 ); 4 ( ) ( 1 , 2 , 3 ) = 4 ( ) ( 1 , 2 , 3 ) 4 ( ) ( 1 , 2 , 3 ) (11) For including the representation of a fourth-order transfer spectrum while keeping the spectrum comprehensible, basic linear systems are considered. To approximate the dynamic behaviour of common mechanical structures with dominant resonances, Figure 5 shows the transfer function and the corresponding fourth-order transfer spectrum of an ideal bandpass filter. This can be interpreted as approximating the dynamic behaviour of a single-degree-of freedom (SDOF) system with a pronounced resonance. Interestingly, the portions of the fourth-order transfer spectrum indicating transfer of non-Gaussianity through such a structure lie exclusively on the planes . (a) (b)

1

bandpass

4 ( ) [−]

H xy [-] | | 0

SDOF

f [Hz]

f Ny

Figure 5: (a) transfer function (b) 4 th -order transfer spectrum for narrow bandpass