310 M. Misawa and H. Kawasoe Frequency error e1 , % 5 4 3 2 1 -10 0 10 20 30 40 Identified, comp.1 Original, comp.1 Identified, comp.2 Original, comp.2 Mode number Frequency error e2 , % 5 4 3 2 1 -5 0 5 10 15 Identified with 3 modes Identified with 7 modes Original Mode number a b Fig. 29.6 Frequency error. (a) "1. (b) "2 1 3 5 7 9 1113151719 -1 0 1 Location number Amplitude Exact Test 1st mode 1 3 5 7 9 1113151719 -1 0 1 Location number Amplitude Exact Test 2nd mode Fig. 29.7 Mode shape of cantilever beam error of the first mode is 0, this does not mean that the first frequency is accurate. The first simulated test frequency equals the first analysis frequency, but both frequencies include modeling errors. If modeling errors in original mass and stiffness matrices are small, the second and higher frequencies ought to be accurate. Because original frequency error is not small, it is difficult to consider the simulated test frequency as beam frequency. By identifying mass and stiffness matrices of tested components with the lower three modes, frequency error "1 is significantly reduced (maximum error 0.5 %) for both tested components. When using seven modes for system identification, the component modal tests provide the simulated test frequency identical with identified frequency. This suggests that one must check the influence of the number of modes used on identified frequencies because error of identified frequency affects frequency error "1. Figure 29.6 (2) shows frequency error "2. If mass and stiffness matrices have no modeling errors, predicted frequencies take the same value. Therefore, frequency error "2 is an indication to predict accurate target frequencies. The maximum frequency error of "2 is 0.5 %. Because frequency errors "1 and"2 are very small, we conclude that the predicted frequencies are beam frequencies. Figure 29.7 shows examples of comparison between exact and simulated test modes of the beam. The solid line shows the exact mode, and the dotted line indicates the test mode obtained by simulated component modal tests. It can be seen from Fig. 29.7 that the simulated test modes are similar with the exact modes for both the first and third modes. Table 29.9 lists MAC representing the correspondence between the exact and test modes. MAC is higher than 0.9. For the second, fourth and fifth modes, MAC is also higher than 0.9. This means that the test modes are almost identical to the exact modes.
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