314 M. Brehm and A. Deraemaeker 10−5 10−4 10−3 10−2 10−1 100 0 100 200 300 400 500 600 V(Re(F¯gf ∗ )) m2s2 circular frequency rad s linear combiner 1 corresponding row number of covariance matrix 10−7 10−6 10−5 10−4 10−3 0 100 200 300 400 500 600 1 25 50 75 100 101 125 150 175 200 V(Re(F¯gf ∗ )) m2s2 circular frequency rad s linear combiner 2 corresponding row number of covariance matrix approach 1 approach2 approach3 a b Fig. 30.2 Variance of the real part of the discrete Fourier transform of the first and second linear combiner (lc) using a rectangular window with compact support between 8 and 9 s approach1 1 100 200 300 400 column number of correlation matrix 1 100 200 300 400 row number of correlation matrix real lc1 real lc2 imag lc1 imag lc2 imag lc2 imag lc1 real lc2 real lc1 approach2 1 100 200 300 400 column number of correlation matrix 1 100 200 300 400 row number of correlation matrix real lc1 real lc2 imag lc1 imag lc2 imag lc2 imag lc1 real lc2 real lc1 approach3 1 100 200 300 400 column number of correlation matrix 1 100 200 300 400 row number of correlation matrix -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 real lc1 real lc2 imag lc1 imag lc2 imag lc2 imag lc1 real lc2 real lc1 a b c Fig. 30.3 Correlation matrix for the real and imaginary (imag) parts of the discrete Fourier transforms of both investigated linear combiners (lc) 30.3.2 General Example Each approach detailed in Sect. 30.2 has been applied with a time step t D2 11 s (corresponding to a sampling frequency of 2,048 Hz) to calculate the statistics of the discrete Fourier transform of the two linear combinations. Due to the design of the modal filters, the variances of the discrete Fourier transforms of the first linear combiner (lc1) have peaks approximately at the first and second circular peak frequency and the variances related to the second linear combiner (lc2) have peaks approximately at the second and third circular peak frequency. As real and imaginary parts of the variances showed an almost identical behavior over the frequency range, only the real part of the discrete Fourier transforms of the two linear combiners are depicted in Fig. 30.2 between 0 and 622.03rad s . The presented frequency range embraces the first 100 discrete frequency steps. The remaining frequency steps until 2 2 t D 6433:98 rad s were not of interest in this study and were disregarded in the calculations and figures. The steady state for this system was reached after about 8 s. Therefore, a rectangular window with a compact support between 8 and 9 s has been applied to create a time frame with a circular frequency step ! D 2 T D 2 rad s . The corresponding correlations are presented in Fig. 30.3. For the sample-based approach 3, 10,000 independent sample sets of excitations have been applied.
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