8 Iterative Pole-Zero Model Updating Using Multiple Frequency Response Functions 69 102 103 104 102 103 104 102 103 104 −200 −100 0 −200 −100 0 −200 −100 0 Frequency [Hz] Gr Ge Gu r G1 [m/N] indB G2 [m/N] indB G3 [m/N] indB Fig. 8.2 Original, experimental, and updated G1 (top), G2 (middle), and G3 (bottom) 0 1 2 3 4 5 6 7 8 2 4 6 8 log10( i) Iteration x 105 Fig. 8.3 log10. i/ for IPZ model updating using G1 and G2 It is assumed that Ge;1 and Ge;2 are available and that the frequency range of interest is Œ40;5000 Hz. Therefore, the original model Gn will be reduced to Gr considering the first five eigenmodes plus one residual flexibility mode for the truncated modes (modes: 6; : : :;24). A coordinate transformation is applied so that the reduced model is expressed in terms of the desired DOFs qr D Œw4; w6; w8; w12; w16; w20 ; containing the actuator, sensors, and the unmeasured performance variables DOFs. The goal is to match the poles and the zeros of the reduced-order Gr;1 and Gr;2 with the poles and the zeros of the experimental Ge;1 andGe;2, respectively, in the frequency range of interest. According to Fig. 8.2, the first four poles of Ge;1 andGe;2, the first three zeros of Ge;1, and the first two zeros of Ge;2 lay within the frequency range of interest, hence are assumed to be extracted from the measurements. The first four poles of Gr;1 and Gr;2 are updated using the first four (real) eigenvalues of the reduced-order stiffness matrixKr, i.e. Œ p;1; : : :; p;4 . Furthermore, the first three zeros of Gr;1 and the first two zeros of Gr;2 are updated using the (real) eigenvalues Œ z1;2; : : :; z1;4 and Œ z2;3; z2;4 of the substructure reduced-order stiffness matrices Ks1 and Ks2, respectively. Using the discussed IPZ model updating algorithm withı D10 5, the algorithm converged to i D244:58ini D755;634 iterations. Although many iterations are needed, the required calculation time is limited because a reduced-order model is used. Note that the number of iterations also depends on the value of the stop criterion parameter ı. It can be seen from Fig. 8.3 that i decreased rapidly during the first iterations, but it takes many iterations to fulfill the convergence criterion of (8.13) with ı D10 5. Convergence of the IPZ model updating toward a (local) minima of (8.7), results in the updated reduced model Gu composed of Mr, Br, andKu r;s2 . Figure 8.2 shows that Gu;1 andGu;2 match very well in terms of both poles and zeros with the poles and the zeros of Ge;1 and Ge;2 in the frequency range of interest. Moreover, Gu;3 which resembles the unmeasured performance variable also matches very well in terms of both the poles and the zeros with those fromGe;3.
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