282 Q. Fei and D. Jiang Table 27.2 Lower and upper bounds for modal frequencies: Incorrect initial model Mode order Experimental modal frequency (Hz) Lower bound (Hz) Upper bound (Hz) Initial modal frequency (Hz) Modal frequency (EI D4,560Nm2) 1 37:5 36:5 47:1 41:9 42:1 2 236:8 228:7 295:3 262:4 264:1 3 657:6 640:6 827:0 734:8 739:7 4 1;256:3 1;256:2 1;621:8 1;441:0 1;450:6 5 1;996:6 2;079:9 2;685:1 2;385:8 2;401:6 6 2;889:7 3;115:9 4;022:6 3;574:2 3;597:9 EI Kt Kr 700mm Fig. 27.3 Correct initial model with elastic-end beam Table 27.3 Lower and upper bounds for modal frequencies: Correct initial model Mode order Experimental modal frequency (Hz) Lower bound (Hz) Upper bound (Hz) Initial modal frequency (Hz) 1 37:5 28:3 43:7 33:9 2 236:8 187:5 273:9 220:6 3 657:6 522:1 760:9 613:9 4 1;256:3 974:9 1;464:2 1;155:0 5 1;996:6 1;538:2 2;350:5 1;819:8 6 2;889:7 2;297:4 3;397:8 2;687:1 From the second column to the forth column of Table 27.2, it could be found that the fifth and the sixth experimental modal frequencies are not within the bounds. This implies that it’s impossible to find a certain groups of parameter values in the given closed interval 3;420 Nm2 5;700 Nm2 that could minimize the deviations between the experimental and the computational modal frequencies of these two modes using the given the model configuration. According to criterion 1, with this given definition domain for the selected parameter, the model is not updatable. For another point of view, modal frequencies of the beam computed using fixed-end configuration and correct parameter value (4,560 Nm2) are listed in the sixth column. It is obvious that the computed modal frequencies differ much from the experimental data even if there is no parameter error. 27.3.3 Evaluation of Initial Model with Correct Boundary Condition In this case, the beam is modeled correctly using the spring elements to model the elastic boundary condition. Figure 27.3 shows the initial model of elastic-end beam. The initial model also consists of ten beam elements. The value of EI is set to be 4,500 Nm2. The values of spring stiffness are set to be Kt D2.0 10 7 Nm 1 and Kr D5.0 10 4 Nm/rad. And the definition domain for EI is set to be 3;420 Nm2 5;700 Nm2 . The definition domain of spring stiffness are set to be 0.3 and 2 times those of the ‘real’ values given in Sect. 27.4.1, they are 1:2 107 Nm 1 8 107 Nm 1 and 3 104 Nm 1 2 105 Nm 1 respectively. The range of definition domain of spring stiffness exceeds the range of EI because the uncertainties in boundary conditions are general larger than the uncertainty in EI. Modal frequencies of the initial model are listed in the fifth column of Table 27.3. The lower and upper bound for the modal frequencies are listed in the third and forth column of Table 27.3. From the second column to the forth column of Table 27.3, it could be found that all of the six experimental modal frequencies are within the lower bound and the upper bound. It means that the initial model could be updated to minimize
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