2.2.4 Summary In summary, one has three formulations that allow us to determine matrix B jj H : First formulation 1 B A A A A C A A A A jj ji ij ji ii ii ij ji ij jj H HHHH HH HH H (16) Second formulation 1 B A A C A jj jj jj jj jj jj H H H H I H (18) Third formulation -1 B A A A A C A A A A jj jj jj ji ij ij jj ji ij jj H HHHH HH HH H (25) 3 SIMULATION STUDIES To illustrate the performance of the three formulations one uses the beam of figure 2, constituted by three components, A1, B and A2, forming structure C; A1 and A2 form sub-structure A. Fig. 2 Test item The aim is to characterize component B (our “joint”), evaluating BH , assuming that AH is calculated analytically and CH is calculated through experiments (simulated, in this case). Using the finite element method with beam elements with four degrees of freedom, each component is divided into eight elements and considering only the nodes shown in figure 2 as coordinates i and j where it is possible to measure and excite the structure. The characteristics of each component are displayed in table 1. Table 1 Characteristics of the components of the beam Beam Length Width Thickness ɋ E ȡ A1 300 mm 25 mm 3 mm 0.3 210 GPa 7850 Kg/m3 B 400 mm 25 mm 6 mm 0.3 210 GPa 7850 Kg/m3 A2 300 mm 25 mm 3 mm 0.3 210 GPa 7850 Kg/m3 3.1 Choice of the formulation To simulate the experimental errors, one has imposed a 1% perturbation in matrix CH for the three formulations. The elements 11 and 12 (chosen just for comparison) of matrix B jj H obtained with the three formulations are shown in figure 3. The major disturbances in the responses are due to numerical problems in the matrix inversions in each of the formulations. The variability of the results in figure 3 is quite high. Figure 4 represents the module of the differences between the numerically exact response and the response obtained by the three formulations. From the three results, the second formulation is the one that produces the smallest error throughout the frequency range. B A1 A2 j j i i 3 1 5 7 9 4 2 6 8 10 13 11 15 17 19 14 12 16 18 20 387
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