64 C. Grappasonni et al. 6.5 Frequency Nonlinear Subsystem Identification of the Beam The low level random test was repeated five times to check that the boundary conditions were not changing during the tests. These several non consecutive tests can be used to evaluate the uncertainty related to the modal parameters estimated for the characterization of the system linear behaviour. The first three bending modes of the beam belong to the analysed bandwidth and are summarised in Table 6.2 together with the associated uncertainties. When the excitation level is increased the softening-hardening nonlinearity affects the peak of the frequency response functions at the resonances. Specifically, for excitation levels up to about 12 Nrms the first mode moves towards lower frequencies (symptom of a softening behaviour), but for higher energy regimes this resonance shifts to the right (symptom of a hardening behaviour). Figure 6.5 shows a close up around the first resonance of the frequency response functions at the tip of the main beam for three excitation levels: the lowest level (0.83 Nrms in red), a regime in which the softening behaviour prevails (6.00 Nrms in blue) and the highest level (25.04 Nrms in gray). It can be also noted that the FRFs become more “noisy” when the excitation increases as a further indicator of nonlinear mechanisms acting in the system. The FNSI method is applied to the high-energy test case (25.04 Nrms) in order to estimate the extended FRF matrix, so then the underlying linear FRF matrix of the system and the coefficients of the nonlinear input force acting at the main beam tip. The latter is considered to be completely unknown and generic cubic splines are implemented to fit its behaviour. Figure 6.6 shows the estimated functions, when the frequency samples are processed within the bandwidth [5–600] Hz and several intervals defying the splines are considered, as indicated by the points on each curve. Specifically, in the present study, the range of the analysis, that is ˙1.1 mm, is divided in a number of intervals ranging from two up to eight. All the functions interpolating the restoring force well represent the softening-hardening behaviour within the range of the analysis. The force, in fact, increases with the displacement following the hardening characterization of the nonlinearity except for a Table 6.2 Experimental linear modes of the beam as assessed from random tests at low excitation level (0.83 Nrms) Mode Natural frequency (Hz) Damping ratio (%) 1 31.63 ˙0.09 0.458 ˙0.051 2 147.82 ˙0.03 0.027 ˙0.002 3 407.11 ˙0.04 0.119 ˙0.014 15 20 25 30 35 40 45 50 −100 −95 −90 −85 −80 −75 −70 −65 −60 −55 −50 Frequency (Hz) FRF at main beam tip (dB) 25.04 Nrms 6.00 Nrms 0.83 Nrms Fig. 6.5 Frequency response functions at the tip of the main beam for excitation levels equal to 0.83 Nrms (red solid line), 6.00 Nrms (blue solid line) and 25.04 Nrms (gray solid line) (Color figure online)
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