30 S.E. Obando et al. Fig. 2.14 Time response comparison for expanded QXN model response (in green) and reference unmodified model (in red). Also shown in blue, modified reference model forced response 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time (sec) Displacement Time Domain Response Near Base Reference Mod. Ref New Expanded Nodes 17-20 DOF 33,35,37,39 F Node 10 DOF 19 a basis for the solution of the NDOF space. The DOF selected in this case resulted in a large condition number for the transformation matrix indicating that the projection vectors are not linearly independent and the matrix is ill-conditioned leading to large errors during the numerical computations. Thus, the proper number of modes and a redistribution of DOF can be performed to yield a well-conditioned matrix and mitigate errors in the numerical process. Case B-2.2 will show that a properly selected perturbed transformation matrix can yield accurately prediction of the expansion of the unmodified system response (reference model). 2.4.7.2 Case B-2.2 Reference cantilever beam response at nodes 4, 9, 12, 16, and 20 was extracted from the full space solution. The nodes were selected from locations that properly captured the mode shapes of interest and gave a fair distribution of points along the length of the beam. The key point is that the DOF selected lead to well-conditioned and fully ranked transformation matrices and therefore a set of projection mode shape vectors that fully span the space of the system. The forced response was once again expanded to NDOF using the new transformation matrix h b TU i obtained from the SEREP reduction of the modified beam at the new ADOF space. The DOF locations and the TRAC are shown in Fig. 2.15. The full NDOF space was accurately predicted as shown in the TRAC of Fig. 2.15. Note that the DOFs near the base of the beam which were previously inaccurate (Fig. 2.14) are now approximated well in the expansion process even though the mode shapes are inexact representations of the unmodified model. Figure 2.16 shows the time response near the base of the beam for the new ADOF set. As highlighted in this and the previous case (B-2.1), the DOF selection plays an important role in the accuracy of the predicted response. While the carefully selected 5 DOF used in this case produced an accurate full space prediction of the response of the reference model, similar results can be obtained with a less than ideal DOF selection. A uniform distribution of points along the length of the beam works well for the model considered here. However, real world structures with changes in material properties and cross section along the length can lead to large concentrations of DOF at sections of the system where modal information is more abundant. The most important aspect lies in obtaining a transformation matrix that spans
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