In view of the qualitative value of this approach, we note that the objective followed here is to derive a simplified realistic model in order to illustrate the numerical computation procedure of NNMs. Accordingly, in the present study, the nonlinear behavior is modeled by adding negative cubic stiffness nonlinearities into the linear part of the connections. An indicative value of −1013N/m3 is adopted for each connection. Finally, the nonlinear system is then constructed from the reduced-order model by means of cubic springs positioned vertically between both corresponding nodes retained on either side of connections. 4.2 Nonlinear Normal Modes From the nonlinear reduced-order model, the numerical computation of NNMs is realized in the MATLAB environment using the previously developed algorithm. The goal followed here is to show that the proposed method can deal with complex structures such as this real-aircraft model. In this context, this section focuses on some specific modes. 4.2.1 FUNDAMENTAL NNMS The modes of the aircraft can be classified into two categories, depending on whether they correspond to wing motions or not. The modes localized mainly on other structural parts (such as the vertical stabilizer, the horizontal tail or the fuselage) are almost unaffected by the nonlinear connections located at the wing tips. Only the modes involving wing deformations are perceptibly affected by nonlinearity. According to the relative motion of the fuel tanks, these modes are more or less altered for increasing energy levels. An unaffected mode is first examined in Figure 8. It corresponds to the nonlinear extension of the first tail bending LNM (mode 13 in Table 2). In this figure, the computed backbone and related NNM motions are depicted in the FEP. The modal shapes are given in terms of the initial displacements (with zero initial velocities assumed) that realize the NNM motion. It clearly confirms that the modal shape and the oscillation frequency remain practically unchanged with the energy in the system. Modes involving wing deformations are now investigated. The first wing bending mode (i.e., the nonlinear extension of mode 10 in Table 2) is illustrated in Figure 9. The FEP reveals that this mode is weakly affected by the nonlinearities. The frequency of the NNM motions on the backbone slightly decreases with increasing energy levels, which results from the softening characteristic of the nonlinearity. Regarding the modal shapes, they are almost similar over the energy range and resemble the corresponding LNM. MAC value between the NNM shapes at low and high energy levels (see (a) and (b) in Figure 9) is 0.99. Figure 10 represents the FEP of the first (symmetric) wing torsional mode (i.e., mode 19 in Table 2). For this mode, the relative motion of the fuel tanks is more important, which enhances the nonlinear effect of the connections. As a result, the oscillation frequency have a more marked energy dependence along the backbone branch. On the other hand, the modal shapes are still weakly altered. MAC value between the NNM shapes on the backbone at low and high energy levels (see (a) and (b) in Figure 10) is equal to0.98. In addition, the FEP highlights the presence of three tongues, revealing the existence of internal resonances between this symmetric torsional mode and other modes. These observed modal interactions are discussed in the next section. Finally, the second (anti-symmetric) wing torsional mode (i.e., mode 20 in Table 2) is plotted in the FEP of Figure 11. While the oscillation frequency is noticeably altered by nonlinearity, modal shapes are again slightly changed. Over the energy range of interest, the decrease in frequency is around5%along the backbone branch. MAC value between the modal shapes at low and high energy levels (see (a) and (b) in Figure 11) is 0.97. It shows that the nonlinearities spatially localized between the wing tips and the tanks weakly influence the NNM spatial shapes. Besides the NNM backbone, one tongue is present at higher energy. For information, the computation of the backbone branch up to the tongue needs 20 min with 100 time steps over the half period (using Intel i7 920 2.67GHz processor). Due to the presence of turning points, the computation of the tongue is more expensive and demands about one hour. Similar dynamics were observed for the higher modes and are not further described herein. 4.2.2 INTERNALLY RESONANT NNMS Besides the backbone branches, the previous FEPs show the presence of tongues of internally resonant NNMs. Following a resonance scenario similar to that described in [3], theses additional branches emanate from the backbone of a specific NNM 235
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