of the response channel, also as shown in Fig. 19.2, in which we clearly see the serious problem of modal interference among modes of this 2-DOF system. For the original ITD method, if a system has relatively heavy damping, it becomes difficult to accurately identify the damping ratios and the mode shapes, due to the interference among modes of the system. In general, the original ITD method can yield reasonably accurate results only when modal damping ratios of the system are below10%. According to the theory presented in the previous sections, we have proposed a theoretical modification for ITD method, and extended the ITD method for modal identification of over-damped structural systems. In conjunction with the proposed assurance index, we can sorted the eigenvectors of the system matrix used in extended ITD method corresponding to the vibrating modes of a structural system. Therefore, the extended ITD method could then be applied to identify modal parameters of an over-damped system. The evaluation results of assurance index kij between the eigenvectors, φi and φj, of the system matrix are summarized in Table 19.1, which shows that the eigenvectors, φ1 andφ4, correspond to the vibrating modes of a structural system in one pair (k14 ¼k41 ¼0.8120), as well as the other eigenvectors, φ2 and φ3, correspond to the vibrating modes of a structural system in another pair (k23 ¼k32 ¼0.9326). The results of modal parameter identification are summarized in Table 19.2, which shows that the errors in natural frequencies are less than 2 % and the error in damping ratios is less than 1 %. Furthermore, to keep track of the target modes, we utilize the MAC (Modal Assurance Criterion), that has been extensively used in the experimental modal analysis. The definition of MAC is [9] MAC ΦiA; ΦjX ¼ ΦiA f g T ΦjX ∗ 2 ΦiA f g T ΦiA f g ∗ ΦjX T ΦjX ∗ , ð 19:9Þ where ΦiA and ΦjX represent two mode shape vectors of interest, and the superscript * denotes the complex conjugate. The value of MAC varies between 0 and 1. When the MAC value is equal to 1, the two vectors ΦiAandΦjXrepresent exactly the same mode shape. Observing the MAC values, which signify the consistency between the identified and the theoretical mode shapes, we found all modes are identified accurately (MAC 0.9). 19.5 Conclusions The conventional Ibrahim time-domain method (ITD) using free-decay responses of structures has been extensively used in the modal-identification analysis, however, which is only applicable to identify the modal parameters of an under-damped structure. In this paper, we have proposed a theoretical modification for ITD method, and extended the ITD method for modal identification of over-damped structural systems. Furthermore, it is shown that the eigenvectors of the system matrix through the extended ITD method corresponding to the vibrating modes of a structural system in pair are sorted using the propose assurance index. Numerical simulations confirm the validity of the proposed method for modal identification of over-damped structural systems. Table 19.1 Evaluation results of assurance index kij between φi and φj i j 1 2 3 4 1 1 0.0593 0.0489 0.8120 2 0.0593 1 0.9326 0.4176 3 0.0489 0.9326 1 0.4372 4 0.8120 0.4176 0.4372 1 Table 19.2 Results of modal parameter identification of the 2-dof over-damped system Mode Natural frequency (rad/s) Damping ratio (%) MAC Exact ITD Error (%) Exact ITD Error (%) 1 9.09 9.24 1.65 120.00 120.78 0.65 0.99 2 13.13 12.93 1.49 120.00 119.47 0.44 0.99 19 Modal Identification of Over-Damped Structural Systems Using Extended Ibrahim Time-Domain Method 147
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