## 3 The Carrier-Launcher System Performance: Preliminary Analyses and Results
### Modal Analyses and Aeroelastic Analyses of the Carrier-Launcher System  
As a major issue in the development of such lifting architectures is related to aeroelastic problems, because of the initial sizing processes, the last part of the research involved a preliminary study of the aeroelastic behaviour of the flying wing. To conduct this study, the finite element software MSC NASTRAN was used [57]. In this conceptual design phase, a half-wing model was studied. A two-spar wing-box structure was modelled, making standard assumptions in terms of sizing of the various components, and an aluminium alloy was selected as the structural material [30]. Both ribs and spars (Table 8) are tapered, and they were modelled with shell elements. The landing gears have been modelled as concentrated masses positioned in the correct stations, while the launcher and the two engines are connected to the structure with the RBE2 connection of NASTRAN; the launcher is placed in the root section (half of its inertia characteristics have been introduced). Tables 9 and 10 contain the data used to prepare the finite element model. The structural mesh of the wing-box is composed of 4165 nodes and about 4446 CQUAD4 elements. Figures 16 and 17 show, respectively, a sketch of the geometry of the half-wing model and a sketch of the structural mesh implemented in PATRAN [58]. In NASTRAN (_Fight Loads_ tool), the aerodynamic surface covers the entire plan form of the wing and is connected (through surface splines) to the displacements of the structural nodes of the wing-box upper skin. As for as the aerogrid, it is built by fragmenting the surfaces into panels; each panel is numbered and perfectly defined. We built two aerogrids: one for the inboard surface and one for the outboard surface in a sequential way along the span. Figure 18 shows an example of the aerodynamic mesh which can be implemented to carry out the aeroelastic analyses with the software NASTRAN [57]. In this case, we used five subdivisions in the wing chord direction and a total of ten subdivisions in the wingspan direction. Two types of modal analysis were performed: the first with the root section free to move in the vertical plane (in this case only the longitudinal translation is constrained for all the nodes of the root section), and the second with the root section clamped. In the first case (free wing), for all the nodes of the root section, the degrees of freedom \"1345\" are set equal to zero (the two translations in the chord direction UX and in the wingspan direction UZ, and the two rotations around the chord direction ROTX--rolling--and the in-plane rotation ROTY--yaw). Vertical translation (UY) and pitching degrees of freedom (ROTZ) are free. In the second case (clamped wing), all nodes at the root section have been fixed; therefore, all degrees of freedom \"123456\" are set to zero (in a Global Reference Frame). Table 11 shows the list and the comparison of first natural frequencies calculated with the mentioned modal analyses of the free half-wing and clamped half-wing, respectively. Figure 17: Sketch of Half-Wing structural mesh prepared with PATRAN [30]. Figure 18: Example of aerodynamic mesh and panel numbering in NASTRAN [30] (circled numbers indicate nodes of the aerogrid). Table 12 contains a comparison of natural modes of vibration in terms of their qualitative shape. As is known, due to the different constraint conditions, the free-wing model has slightly higher frequency values. The aeroelastic analyses were performed at sea level and in cruise flight condition. In these calculations, we assumed \(h=11,000\) m as cruising altitude. To obtain preliminary results, the half-wing clamped model was considered (neglecting rigid modes effect). We selected the \(p\)-\(k\) method [57] to extract the complex eigenvalues (10 complex roots). A structural damping of 0.02 was assumed.