230 S. Vettori et al. Table 23.1 Three DOFs system mode sets MIMOset VDP reconstructed set Mode type Theoretical frequency (Hz) Undamped frequency (Hz) Damping (%) Undamped frequency (Hz) Damping (%) 1 S 54,476 54,476 1.03 54,481 1.12 2 A 100,658 100,658 1.90 100,651 1.91 3 S 131,517 131,517 2.48 131,514 2.49 S symmetric, Aantisymmetric a b Fig. 23.3 Airplane mockup. (a) Airplane mockup experimental setup. (b) Actual and virtual driving points FRFs Figure 23.2b compares the actual driving point FRFs, i.e. obtained by using the conventional method, and the virtual driving points FRFs obtained by applying the new technique. The black and the green curve are the actual driving points FRFs: they are perfectly overlapped thanks to the symmetry of the system. These two curves both show the three peaks correspondent to the three mode shapes of the three DOFs system. The red curve is the virtual driving point FRF obtained when applying the VDP method using data belonging to the symmetric sweep: the peaks in the FRF are only two and they are expected to be related to symmetric mode shapes. The blue curve instead, is the virtual driving point FRF obtained when the method is applied to the antisymmetric sweep data: the peak is only one and it is expected to be related to an antisymmetric mode shape. The mode sets extracted from the FRFs shown in Fig. 23.2b are compared in Table 23.1 in terms of natural frequencies and damping ratios. Moreover, the identified symmetric or antisymmetric nature of each mode is reported. It is possible to notice that the VDP method correctly identifies the mode set of the system and that the symmetric and antisymmetric nature of the modes actually corresponds to the expected one: the first and third modes are symmetric, while the second one is antisymmetric. 23.4.2 Airplane Mockup: Polymax Stabilization Diagrams and Mode Shapes The airplane mockup shown in Fig. 23.3a has been tested at Siemens Industry Software laboratories exciting it through two shakers placed at the wings tips and measuring its responses through 18 mono-axial and 6 tri-axial accelerometers. The data have been acquired exciting the system with a logarithmic sine sweep at 1 oct/min, symmetric during a first run and antisymmetric during the second, in a frequency band between 5 and 110 Hz. Figure 23.3b illustrates the actual and the virtual driving points FRFs on the same plot. The virtual driving point FRF obtained from the symmetric sweep (red curve) shows only some peaks, which are related to the symmetric modes of the system. Indeed, the red curve detects the first peak of the actual driving points FRFs, which is associated to the first wings bending of the mockup. The virtual driving point FRF obtained from the antisymmetric sweep (green curve) instead, does not show the mentioned first peak, because it only identifies peaks associated to antisymmetric modes. An interesting advantage of the VDP method lies in the clearness of the Polymax stabilization diagrams, as shown in Fig. 23.4, if compared with the ones obtained by using the conventional method.
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