308 M. Misawa and H. Kawasoe 1 2 3 4 5 6 7 8 1.0 1.1 1.2 1.3 1.4 Element number Modification coefficient Exact Group 1 Group 2 Group 3 Group 4 9 101112131415161718 1.1 1.2 1.3 1.4 1.5 Exact Group 1 Group 2 Group 3 Group 4 Element number Modification coefficient a b Fig. 29.3 Modification coefficient for mass matrix identification. (a) Component 1. (b) Component 2 1 2 3 4 5 6 7 8 0.6 0.8 1.0 1.2 1.4 Element number Modification coefficient Exact Group 1 Group 2 Group 3 Group 4 9 101112131415161718 0.6 0.8 1.0 1.2 1.4 Element number Modification coefficient Exact Group 1 Group 2 Group 3 Group 4 a b Fig. 29.4 Modification coefficient for stiffness matrix identification. (a) Component 1. (b) Component 2 concerning the Y bending mode does not change as element gets close to the tip of the beam. Modification coefficient of group 3 concerning the Z bending mode approaches the exact values, but it does not change as the element approaches the tip of the beam. For component 2, modification coefficients of groups 2 and 3 change a little near the free edges. Modal stiffness can provide the reason [26]. For example, modal stiffness of the first mode is very small at elements near the tip of the beam because the modal stiffness constraints are satisfied by correcting the grouped matrices of elements near the root of the beam. Because the number of modes used for the identification is small, these modification coefficients do not approach the exact value. Therefore, it is expected that the modification coefficient will be improved by increasing the number of modes. The modification coefficients are found when using the lower seven test frequencies and modes: two Y bending modes and five Z bending modes. Modification coefficient of group 3 is almost identical with the exact value. Substituting the modification coefficients and for Eq. (29.14) provides the identified mass and stiffness matrices. The residues of the mass property constraints and the mode orthogonal constraints were calculated to confirm modeling error reduction of mass matrix. The maximum residue of the constraints is almost zero. Also for stiffness matrix identification, the maximum residue of the modal stiffness and the mode orthogonality constraints is almost zero. Natural frequencies and modes computed with the identified mass and stiffness matrices agree well with exact ones of each tested component. The mass and stiffness matrices of the beam are assembled with the identified mass and stiffness matrices of tested components. Namely, the dynamic equation of the beam can be obtained. Table 29.7 shows natural frequencies of the beam obtained by solving this equation. We call these frequencies “identified frequencies” hereafter. Arabic numbers in parenthesis show frequency error to exact frequency. As the number of test frequencies and modes used in system identification has an influence on identified mass and stiffness matrices, the lower three and seven modes are used in system identification to confirm the effect of the number of modes on natural frequencies of the beam. Table 29.7 shows that frequency error decreases with increasing the number of modes. Identified frequencies are almost identical with the exact frequencies. Exact additional mass and stiffness are calculated with exact frequencies shown in Table 29.2. Analysis frequencies and identified frequencies are used to find original and identified additional mass and stiffness, respectively. Errors in additional mass and stiffness of component 1 are shown in Fig. 29.5 when they are attached to the selected coordinates listed in
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