786 S. Rahrovani et al. k1 c1 m1 m2 m3 c2 c3 c4 k2 k3 k4 X1 X2 F X3 Fig. 73.1 A3-DOF mass-spring-damper system 60 40 20 0 -20 -40 -60 -80 -100 -120 -140 100 101 102 Magnitude (dB) Frequency (rad/sec) full model L2 sense approximation H2 sense approximation HSV guided approximation Fig. 73.2 Frequency response comparison of second order approximants via different approaches Table 73.1 Modal ranking based upon different dominancy metrics Dominancy metric values Decision criteria First mode Second mode Third mode H2 guided NQE 1,308 437 825 L2 guided NQe 1,308 437 825 HSV guided HSV 14,959 8,418 20,662 examples. Specifically these examples investigate (i) the optimality of the solution, (ii) The performance and error analysis of the method and (iii) the effect of multiple eigenvalues on metric non-uniqueness. 73.3.2 Performance and Error Analysis In this section a complete performance and error analysis of the proposed modal dominancy metric is carried out. The performance of the proposed method for modal reduction of lightly damped structures is validated and the results are compared with other dominancy metrics. Furthermore, the study attempts to unify the H2 and H1 model-reduction performance objectives such that the reduced order model satisfying an optimized H2 cost bound as well as a pre-specified H1bound. Details are demonstrated via the following illustrative case study. Example 1. Consider the SISO structure demonstrated in Fig. 73.1. In this case the damping values are chosen such that all structural resonances become lightly damped. This can be seen in the transfer function of the system shown in Fig. 73.2. The results of the modal rankings based on different metrics are shown in Table 73.1. Different conclusions can be drawn from the results. First, according to Parseval’s theorem, it should not be surprising that for such a lightly damped system the SkeltonL2 metric and theH2 metric proposed here lead to identical modal contribution weightings. This is due to the fact that both metrics have used the same performance criterion in quadratic sense, one derived in frequency domain and one in the time domain, kGk H2 Dk gk L2 (73.29)
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