312 M. Brehm and A. Deraemaeker By means of a reordering of the frequency response function values H.!k//i;j for all discrete frequencies !k for k D 1;2; : : : ; mw 2 C1, a certain excitation degree-of-freedomj and linear combiner i, amatrix Pi;j Dh.H.!1//i;j .H.!2//i;j : : : .H.!mw 2 C1//i;j i T (30.23) can be defined withPi;j 2 C mw 2 C1. The discrete Fourier transforms of the linear combiner for all circular frequency steps are described by QFgwi D mf X jD1 Pi;j ˝11 mw ıB qwfj D mf X jD1 Z2i;jfj (30.24) and QFgwi .!k/ 2 C mw 2 C1. Finally, the discrete Fourier transforms for all linear combiners i D 1;2; : : : ;mg and excitation degrees of freedom j D1;2; : : : ;mf are described by the random vector and the linear deterministic operator, FNg2 D 2 6 6 6 6 4 QFgw 1 QFgw 2 : : : QFgwmg 3 7 7 7 7 5 and NZ2 D 2 6 6 6 6 4 Z21;1 Z21;2 : : : Z21;mf Z22;1 Z22;2 : : : Z22;mf : : : : : : : : : : : : Z2mg;1 Z2mg;2 : : : Z2mg;mf 3 7 7 7 7 5 ; (30.25) respectively. This allows defining the linear relationship FNg2 D NZ2 Nf with NZ2 2 C. mw 2 C1/mg Nmf (30.26) between the discrete Fourier transform of the linear combiners and the excitationNf in the time domain. The vector Nf is defined in Eq. (30.3). In general, the generation of matrix NZ2 is computationally less demanding than the generation of NZ1, because NZ1 involves the computation of the discrete time convolution integral. 30.2.4 Approach 3: Sample-Based Uncertainty Propagation The third approach is a standard sample-based strategy based on a Latin hypercube sampling scheme [10]. The samples are generated from the multivariate distributions of excitationNf defined in Eq. (30.3). For each sample set a modal superposition [1] is performed for the structural system and the resulting single-degree-of-freedom systems are solved by a time integration method according to [9], which is a special solver for the standard Duhamel integral. The linear combination matrixAand the window function ware applied to the obtained response to receive the linear combiners in the time domain. Subsequently, a fast Fourier transformation (FFT) algorithm is applied to derive the discrete Fourier transforms. The discrete Fourier transforms of all linear combiners are assembled according to the left hand side of Eq. (30.16). This assembled complex vector is then separated in real and imaginary parts subsequently placed on top of each other. By performing a sample statistics on the resulting vector, the sample mean value vector and the sample covariance matrix are computed, which are direct estimators of the mean value vector and covariance matrix of the left hand side of Eqs. (30.4) and (30.5), respectively. In general, the accuracy of these estimators can be improved by increasing the number of sample sets. 30.3 Benchmark Study: Three-Degree-of-Freedom System 30.3.1 System Description A three-degree-of-freedom system is considered with the mass matrix and stiffness matrix MD 2 4 0:927 0:000 0:000 0:000 1:617 0:000 0:000 0:000 2:612 3 5 kg and KD 2 4 200000 40000 120000 40000 120000 40000 120000 40000 200000 3 5 N m ;
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