software, was converted and exploited in the Samcef finite element environment for this study. The natural frequencies of the underlying linear system in the [0-50Hz] frequency range are given in Table 2. The first nine modes correspond to aircraft rigid-body modes: six modes are modes of suspensions of the landing gear while the three others are associated to rigid-body motions of the control surfaces (i.e., the ailerons, elevator and rudder). The frequency range of the rigid-body modes is comprised between 0.09 and 3.57 Hz, i.e., noticeably lower than the first flexible mode located at 8.19 Hz. The modal shapes of different elastic normal modes of vibrations are depicted in Figure 5. Figure 5(a) represents the first wing bending mode. The first and second wing torsional modes are depicted in Figures 5(b) and 5(c). These two torsional modes correspond to symmetric and anti-symmetric wing motions, respectively. As shown thereafter, these modes are of particular interest in nonlinear regime since there is a significant deformation of the connections between the wings and fuel tanks. Indeed, the other modes mainly concern the aircraft tail and are consequently almost unaffected by these nonlinear connections. 4.1.2 REDUCED-ORDER MODEL The proposed algorithm for the numerical computation of NNMs is computationally intensive for the large-scale original model possessing more than 80000 DOFs. Since the nonlinearities are spatially localized, condensation of the linear components of the model is an appealing approach for a computationally tractable and efficient calculation. A reduced-order model of the linear finite element system is constructed using the Craig-Bampton (also called component mode) reduction technique [15]. This method consists in describing the system in terms of some retained DOFs and internal vibration modes. By partitioning the complete system in terms of nRremainingxRandnC =n−nRcondensedxC DOFs, thengoverning equations of motion of the global finite element model are written as MRR MRC MCR MCC x¨R x¨C + KRR KRC KCR KCC xR xC = gR 0 (15) The Craig-Bampton method expresses the complete set of initial DOFs in terms of: (i) the remaining DOFs through the static modes (resulting from unit displacements on the remaining DOFs) and (ii) a certain number m<nC of internal vibration modes (relating to the primary structure fixed on the remaining nodes). Mathematically, the reduction is described by relation xR xC = I 0 −K−1 CCKCR Φm xR y =R xR y (16) which defines then×(nR+m) reduction matrixR. yare the modal coordinates of theminternal linear normal modes collected in the nC ×mmatrix Φm = [φ(1) . . .φ(m)]. These modes are solutions of the linear eigenvalue problem corresponding to the system fixed on the remaining nodes KCC −ω 2 (j)MCC φ(j) =0 (17) The reduced model is thus defined by the(nR+m) ×(nR+m) reduced stiffness and mass matrices given by M=R∗MR K=R∗KR (18) where star denotes the transpose operation. After reduction, the system configuration is expressed in terms of the reduced coordinates (i.e., the remaining DOFs and the modal coordinates). The initial DOFs of the complete model are then determined by means of the reduction matrix using relation 16. In order to introduce the nonlinear behavior of the connections between the wings and the tanks, the reduced-order linear model of the aircraft is constructed by keeping one node on both sides of the attachments. For each wing, four nodes are retained: two nodes for the front attachment and two nodes for the rear attachment. In total, only eight nodes of the finite element model are kept to build the reduced model. It is completed by holding the first 500 internal modes of vibrations. Finally, the model is thus reduced to 548 DOFs: 6 DOFs per node (3 translations and 3 rotations) and 1 DOF per internal mode. The reduction is performed using the Samcef software. The generated reduced-order model is next exported in the MATLAB environment. Before proceeding to nonlinear analysis, the accuracy of the reduced-order linear model is assessed. To this end, the linear normal modes of the initial complete finite element model are compared to those predicted by the reduced model. The deviation 230
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