258 M. Mehrkash and E. Santini-Bell log (J) log (J) log (J) log (J) Case 1 Case 2 Case 3 Case 4 Fig. 26.8 Objective functions of the four cases defined in Table 26.2 and 3, modes 5 and 6 are contributed while in cases 2 and 4, the two mentioned modes are ignored. Comparing MAC plots of cases 3 and 4 with cases 1 and 2 demonstrates that the instrumentation used for this test, is not sufficient for capturing modes 10 and 11. This can be verified as in the MAC plots of cases 1 and 2 (which includes modes 10 and 11), large nondiagonal bars can be seen for modes 10 and 11. An interesting point is the role of modes 5 and 6 in the estimation of stiffness parameter. In these two modes, the vibration of beams 14, 15, 17 and 18 (see Fig. 26.6) is considerable. Therefore, it might be required to update the rotational rigidity of these four members too. Therefore, in the next section, the model updating is performed by considering this additional group of stiffness parameters. 26.6.2 Two-Parameter Estimation Analysis In this section, as shown in Fig. 26.10, the connections of beams 14, 15, 17 and 18 to the girders are modeled as semi-rigid joints in addition to the previously introduced partial fixities. The previously defined rotational fixities are considered as group 1 and the new spring constants are put in group 2. Hence, in this analysis there are two unknown stiffness parameters. In Fig. 26.10, the group 1 is shown by red boxes and the group 2 is specified by blue boxes. The contributed modes are the same as the ones in case 3 of the previous section, i.e., modes 1 to 6 and mode 12 are contributed. The analytical, experimental and updated natural frequencies for this analysis are given in Table 26.4. Also, the corresponding MAC plot is shown in Fig. 26.11. Further, the updated values for the two partial fixities are given in Table 26.5. The results of the two-parameter estimation analysis show that considering partial fixities for beam to girder connections does not improve the estimation procedure. In this case, the partial fixity of the parameter 1 converges to the same value of case 3 in one-parameter estimation, while the stiffness of group 2 parameters reaches the upper defined bound. Considering the graphs of Fig. 26.7, this boundary value represents a fixed connection, which was the condition in the one-parameter estimation. Therefore, introducing partial fixities for the beam to girder connections does not provide any information about the semi-rigidity of these joints. Hence, the two-parameter estimation procedure does not show much advantage over the one-parameter estimation.
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