of beam A. A corresponding mode exists in subsystem A, with the exact same natural frequency. That mode was completely unaltered when A was joined to C since it has a node at the connection point, so essentially the same mode exists in both A and C. The zero eigenvalue in ˆBm apparently comes about because this mode’s mass is entirely removed from C by the substructure uncoupling process. Recall from eqs (7-12), that the B system has exactly the same number of modes as the C system, so if a mode is removed completely from C by the uncoupling process then a spurious mode must remain in the model for B. The other two offending eigenvalues are more difficult to interpret. The table shows that the sixth mode of C is the dominant contributor to the first negative eigenvalue, contributing 0.77 of the total value of 1.012. Mode 6, shown with a blue line and open circles in Figure 2, involves axial motion of beam B and bending motion of beam A. As mentioned previously, the lower modes of C all exhibit bending motion of B with beam A undergoing approximately rigid body rotation, so this is the first mode to show significant bending in A. The fact that this mode contributes 0.77 of the 1.012 eigenvalue signifies that this mode carries a significant proportion of the mass associated with bending motion of the transmission simulator, mass which must be removed to accurately predict the natural frequencies of B. Hence, the uncoupling algorithm is working with regard to this mode so long as the amount of mass subtracted is correct. To diagnose the situation further, the orthogonal projection of C’s motion onto the space of A’s modes, ˆxCm , was found using the orthogonal projector Am P in eq. (5) and it is also shown in Figure 2 with a red line and with dots at each of the node points. The zoom view shows that the reconstructed motion matches the true motion very well; the maximum difference between the two is 1.4%. Hence, it seems that this mode’s contribution to CM Q is physical and represents mass that should be removed. -50 0 50 100 150 200 250 300 350 -80 -60 -40 -20 0 20 40 60 80 x (mm) y (mm) -2 0 2 4 6 -40 -30 -20 -10 0 10 20 30 x (mm) y (mm) Figure 2: Shape of Mode 6 of system C: (black/dots) undeformed structure, (blue/circles) mode of C, (red/dots) projection, ˆxCm , of C onto the free modes of A. The other mode shapes that contribute to this eigenvalue were also viewed, revealing that there were significant errors when projecting modes 9 and 12 onto the transmission simulator’s motion; the maximum difference between the actual motion and the projection was 9.4% and 44.6% respectively. Mode 12’s shape is shown in Figure 3 with a blue line. The plot reveals that the 3rd bending mode of the beam would be needed to describe the observed motion, but the model that was used for A only included the first two bending modes (and one axial mode). This reveals that the modal basis of the transmission simulator is inadequate to describe mode 12’s motion. Because the transmission simulator model does not contain the third bending mode, the uncoupling process might erroneously attribute the third-mode motion to other modes, and hence 125
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