66 C. Grappasonni et al. the experimental data in case of the lowest and the highest level of excitation (0.83 Nrms in red and 25.04 Nrms in gray, respectively) with the reconstruction coming from the FNSI method (25.04 Nrms in black). It can be noted that the estimate of the underlying linear FRF as from FNSI matches the experimental FRF at the lowest level of excitation. The frequencyshift of the resonance peak due to the nonlinearity is recovered, although the relative error of 0:97% on the first natural frequency can be highlighted. 6.6 NNM-Based Identification of the Beam Starting from the parameters estimated by FNSI a finite element model of the two beams has been implemented, such that the linear modal parameters were the same of the real structure, assuming the damping matrix to be proportional to the mass and stiffness matrices. The nonlinear force acting at the tip of the main beam has been added to the system as identified by FNSI using cubic splines. In the following the analyses are focused on the first mode which involves the largest displacement of the main beam tip, that is where the geometrical nonlinearity is acting, and hence the strongest effects are addressed. The experimental harmonic forces and accelerations are measured in stationary conditions for varying frequency at different excitation energies. When a 90ı of phase shift is detected between the force and the acceleration, the resonance is occurring since the quadrature condition is fulfilled. At this point an estimate of the nonlinear normal mode can be experimentally achieved and compared with the analogous coming from the numerical evaluation of the backbone from the nonlinear normal mode continuation toolbox. Figure 6.8 shows the experimental harmonic response of the main beam tip for harmonic excitation ranging from 0.01 to 0.88 N. Both up and down sweep directions were tested, but only the former is here represented for the sake of brevity. It can be noted that at first the curves tend to bend to the left and then to the right, as expected by the softening-hardening behaviour already identified in the previous section. Moreover, for high excitations the so-called jump phenomenon appears, that is when suddenly the response amplitude strongly decreases (or increases) for a small increase (or decrease) of the excitation frequency in case of the hardening behaviour. Since the presence of structural damping the resonances are not located at the peak of the responses and these jumps happen after the achievement of the quadrature condition, so at higher frequencies. 24 26 28 30 32 34 36 38 40 42 44 10−2 10−1 100 101 Frequency (Hz) Acceleration (g) Fig. 6.8 Experimental response functions at the main beam tip to several stepped sine excitations at different energies withhighlighted the points where the quadrature condition is fulfilled
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