The design aim is to define suitable configurations where it is possible to see a high number of curve veering and curve crossing phenomena with respect to the angle parameter. The eigenvalues graph shown in Figure 10 represents the expected dynamic behaviour in the frequency range up to 220 Hz. The mode-shapes related to the configuration 0 are used for labelling the curves and for following the eigenfrequencies modifications; different colours are adopted for distinguishing crossing (“C” symbols) and veering phenomena (“V” symbols). MAC index is adopted to follow the mode-shapes and to distinguish crossing from veering. Another result that can be obtained by the FE model is the analysis of modes shapes, especially across the curve veering or curve crossing points. In Figure 11 MAC index between eigenvectors with = 44° and 45° are plotted on the left. They involve the 6th mode (2nd torsional) and the 7th mode (bending); a typical crossing phenomena is evinced. Therefore this case produces a order change of modes, the modes shapes remain the same and the corresponding curves intersect each other. The same approach is used between eigenvectors with = 68° and 69° on the right of Figure 11, where two bending modes are involved in a curve veering phenomena. The 5th and 6th bending modes are very similar first and after occurrence of veering phenomena, in fact the MAC index is not suitable to distinguish two different modes. Increasing the angular resolution between the eigenvector comparison produces always diagonal MAC index, corresponding to not crossing phenomena. Non zeros values of off-diagonal MAC terms are due to non perfect orthogonal properties of eigenvectors. The mass matrix is not proportional to identity because, although beam elements are equal, translational and rotational inertial terms are different and lumped masses are taken into account. Finally, even for the test bench curve crossing and curve veering present the same peculiarity described in the last paragraph. Therefore the design of the test bench allows to validate the theoretical results about crossing and veering and to study, in the future, energy transfer paths. 0 20 40 60 80 0 20 40 60 80 100 120 140 160 180 200 220 Mode shapes for = 0° [deg] Freq [Hz] Mode 1 - 1 Byx Mode 2 - 1 Bzx Mode 3 - 1 T Mode 4 - 2 Bzx Mode 5 - 2 Byx Mode 6 - 2 T Mode 7 - 3 Bzx Mode 8 - 3 T Mode 9 - 4 Bzx Mode 10 - 5 Bzx Figure 10 – Natural frequencies of the structure versus angle configuration . C V C C V C V 333
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