10 Numerical and Experimental Modal Analysis of a Cantilever Beam. . . 111 applied force - P [N] 0 50 100 150 200 250 300 350 400 450 natural frequency - f [Hz] 1B 0 100 200 300 400 500 2B 1T-R 1T-L 3B 2T-R 2T-L 4B experimental with attachment point without attachment point Frequency-loading diagram 1B W(x) Wt (x) Mode shapes 2B 1T-L 1T-R 3B 2T-L 2T-R 4B Fig. 10.3 Comparison of the computed and experimentally measured frequency-loading diagrams for the system with a single attachment point in 0.5L, and the mode shapes for P =300N. The xth beam-dominated mode is marked by “xB”, while the xth tendon-dominated mode intrinsic to the left and right part of the tendon are marked by “xT-L” and “xT-R”, respectively other, only every second tendon-dominated frequency locus of the system with the one attachment point coincides with some frequency locus of the system without the attachment. Although some of the natural frequencies coincide, the mode shapes of these modes are different due to the attachment fixture. The natural frequencies coincide because the length of the tendon’s segment in the system with the attachment is exactly halved, leading to twice as high natural frequencies of any given mode compared to the system with no attachment. The rate of increase of the frequency loci is however the same. Due to a larger number of the tendon-dominated modes for the system without the attachment, more veering regions can be seen. In addition, it can be noticed that the veering regions of the two systems are different. For the system with the attachment point, there is always a frequency locus passing through the veering region while there is no such locus for the system with no attachment. The locus, which passes through the veering regions unaffected, belongs to one of the pair of the tendondominated modes that share the same natural frequency when away from this veering region. This phenomenon can also be observed in the experimental data between 400 N and 500 N at about 400 Hz. From the numerical results, it is possible to determine which tendon-dominated mode (left or right) veers and which passes through the veering region unaffected. However, since the mechanism is not yet theoretically fully understood, it is not further discussed and should be a subject of further investigation. The modes shapes of the system can be seen in the right part of Fig. 10.3. Although the attachment point is not displayed, its position can be deduced from these modes shapes. The first fourth bending modes that are dominated by the motion of the beam (marked as xB) are present. Then, the mode shapes that are intrinsic to the left part of the tendon (marked as xT-L) can be seen. In these modes, the deflection of the beam and the right part of the tendon is minimal and the mode shape is localised to the left part of the tendon. On contrary, the modes of the right part of the tendon (marked as xT-R) are only localised in the right part of the tendon. It should be noted that many more vibration modes exist for very low tendon tensions (e.g. P <50N in the frequently-loading diagram). However, due to the need to provide a reliable starting guess to the numerical solver used and the closeness of the natural frequencies of these modes, it was not possible to map this region comprehensively.
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