Modelling Continuously Morphing Aircraft for ... - Michael I Friswell

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Figure 6: Comparison between FSDPT modal shapes (left) and NASTRAN modal shapes (right). 10 of 23 American Institute of Aeronautics and Astronautics
% Λ LE = 0 ◦ Λ LE = 30 ◦ ε CPT w 8.19 8.6 ε FSDP w 0.85 8.4 ε CPT θ 23.1 26.5 ε FSDP θ 3.9 3.94 Table 2: CPT and FSDP errors relative to NASTRAN results for the wing box example. Λ LE = 0 ◦ Λ LE = 30 ◦ Mode # f NAS [Hz] f FSDPT [Hz] ε [%] f NAS [Hz] f FSDPT [Hz] ε [%] 1 57.87 56.84 1.7 44.19 45.17 -2.2 2 219.95 215.58 1.2 151.17 154.27 -2.05 3 315.32 323.22 -2.5 247.30 254.56 -2.9 4 356.42 370.37 -3.9 342.75 357.1 -4.18 5 705.58 1089 -54.4 596.80 688.07 -15.3 Table 3: Natural frequencies for the wing box example. number of degrees of freedom will be N P ×N q where N P is number of plates and N q is the number of degrees of freedom of each plate. Finally each plate will add to the system a contribution to the kinetic and potential energy that is evaluated with respect to the local coordinate system and thus the formulation obtained in previous sections will be used for this purpose. In the current formulation the displacement compatibility between contiguous plates is imposed by means of stiff springs. In contrast to Demasi, 34 where only longitudinal springs were placed throughout the thickness to impose the cantilever condition and the compatibility, both rotational and longitudinal springs will be used in this work. The displacement compatibility is imposed in global coordinates but expressed in local coordinates as follows: Consider the contiguous plates i and j and let R i and R j be respectively the global to local trasformation matrix of each plate. In order to impose the displacement compatibility of the point A i belonging to body i and the point A j belonging to body j the following expression leads to the potential energy of the springs linking A i and A j ⎡ E = 1 2 (ūi A i − ū j A j ) T ⎢ ⎣ ⎤ k x 0 0 0 0 0 k y 0 0 0 0 0 k z 0 0 ⎥ 0 0 0 k rx 0 ⎦ 0 0 0 0 k ry (ū i A i − ū j A j ) (38) where ū collects the elastic displacements and the rotations of a point of the structure in global coordinates. Thus the global displacement vector can be expressed first in terms of the local displacement ū, via a geometrical rotation matrix, and, consequently, in terms of the Lagrangian coordinates via the matrix of shape functions S. ū = R u = R S q (39) Substituting Eq. (39) in to Eq. (38) leads to the following expression for the potential energy E = 1 2 (qT i K s ii q i + q T j K s jj q j + q T i K s ij q j + q T j K s ji q i ) (40) 11 of 23 American Institute of Aeronautics and Astronautics
Page 1 and 2: AIAA Guidance, Navigation and Contr
Page 3 and 4: Figure 2: Generic trapezoidal wing
Page 5 and 6: II.C. First-Order Shear Deformation
Page 7 and 8: the double integration over the win
Page 9: Vertical displacement [m] (ΛLE = 0
Page 13 and 14: where b ∆T = [α 1 , α 2 , 0] T
Page 15 and 16: 2 x 10−5 Vertical displacement [m
Page 17 and 18: Figure 13: Scheme of the aerodynami
Page 19 and 20: U ∞ 30 m/s α 3 ( ◦ ) ρ 1.22 k
Page 21 and 22: is given in Fig. 18. Moreover, two
Page 23: 22 S. Tizzi, “Numerical Procedure

Modelling Continuously Morphing Aircraft for ... - Michael I Friswell

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