7.2.1 Test case description
For the remainder of this chapter, we consider the 72m-span high-aspect-ratio flying wing model originally created by Patil et al. [8] and subsequently used by Su et al. [7] and Dillsaver et al. [6] , shown in Figure 26. Its properties are shown in Table 8. The airframe has a flat, straight midsection and an outer section with 10◦ dihedral. Three
vertical fins are placed below the midsection and thrust is provided by five propellers mounted forward of the midsection. The payload is placed in the central pod and is variable between 0 (0 %) and 227 kg (100 %).
Figure 26: Configuration of the flying wing [6]. Properties are listed in Table 8.
Elastic/reference axis 25% chord
Aerodynamic centre 25% chord
Centre of gravity 25% chord
GJ 1.65 × 105Nm2 EI2 1.03 × 106Nm2 EI3 1.24 × 107Nm2 m 8.93kg/m I11 4.15 kg m I22 0.69 kg m I33 3.46 kg m Wing clα 2π Wing clδ 1 Wing cd0 0.01 Wing cm0 0.025 Wing cmδ -0.25 Pod clα 5 Pod cd0 0.02 Pod cm0 0
The structural model of the aircraft is created using a separate in-house finite-element beam code [3], with 40 elements for each side of the central section, 20 elements for the outer section and 1 rigid element each for each of the 3 fins under the wing. Eigenvalue analysis was then performed on the structure, obtaining the structural displacement modes and their corresponding eigenvalues defined on the 124 nodes on the airframe. Guyan reduction from a 3D structure is not needed in this case as the airframe is already defined using beams, however the starting point for our analysis is equally the displacement description of the modes and the masses at the nodes. Those are post- processed using the method as before. The payload is considered part of the model, thus varying the payload requires re-computing the modes, it is worth noting that Moulin et al. [144] demonstrated a method in which this is avoided. Applying the method described in Chapter 3 to the modes and frequency information leads to the A, Λ matrices and Γ coefficients. Additional wing section definitions lead to the H coefficients for aerodynamic influences.
The lowest frequency mode shapes in velocity vector (Φ1|123) and the sectional mo-
ment vectors (Φ2|456) in the corresponding force modes are shown in Figure 27. The
angular velocity and sectional force vectors are not shown. Note here that the velocity vector, when plotted in global coordinates, scales directly with displacement vectors. While the definition of force and moment are dependent on the beam direction and the local coordinates of the beam segment.
In this work we are interested in the symmetric response of the airframe and thus de- fine four possible symmetric control actions available on this flying wing: a simultaneous flap deflection by a fixed angle on the entire wing (simultaneous flap δ), a simultaneous change in the thrust in each engine pod (simultaneous thrust FS), a differential flap
deflection by deflecting the flaps on the outboard section (the section with dihedral) in the other direction from the inboard flaps (symmetric differential flap δD), a differential
thrust by increasing output from the two outboard pods and reducing output in the cen- tral pod (symmetric differential thrust FD). The differential flap and thrust inputs are
designed to provide better control on the degree of bending exhibited on the wing. For trimming the airframe however only the simultaneous flap and thrust controls are used with the other two control inputs set to zero. Figure 28 illustrates the flap deflections for symmetric and differential actions.
(a) Velocity vector (b) Moment vector
Figure 27: Field plots of local sectional velocity and sectional moment vectors for the first four structural modes of the 0-payload airframe obtained through the condensation. These corresponds to (from above) first symmetric out-of-plane bending, first antisym- metric out-of-plane bending, first symmetric in-plane bending and first antisymmetric in-plane bending.
7.2.2 Trim Solution Validation
The vehicle, as prescribed by the original work describing it, is flown at 12.2 m/s at sea level and its trim condition is computed for various central pod payloads and for both rigid airframe and the fully flexible airframe. The rigid case uses six rigid body velocity modes only, whereas the flexible case uses 294 symmetric flexible structural velocity modes together with the six rigid body velocity modes selected based on lowest eigenvalue frequency. It was found that such number of modes is required for convergence of the flexible trim solution. Note that the number of force modes (q2) needed is equal to the number of flexible velocity modes, whereas the number of aerodynamic modes (qa) is equal to the total number of velocity modes, multiplied by the total number of
aerodynamic lags (NAE) used to approximate Wagner’s function. As throughout this
work NAE = 2, the rigid model contains 18 modes in total (6 rigid body velocities
and 12 aerodynamic modes associated) and the flexible model contains 1194 structural and aerodynamic modes (NM = 294 velocity and force modes, 6 rigid body and 600
associated aerodynamic modes). Both models contain the same additional T and r states used to integrally track displacements and rotations for each node.
Figure 28: Flap deflections on the flying wing for symmetric and differential actions.
The process of trimming the airframe includes first prescribing an angle of attack α at the centre node of the flying wing, then computing the aerodynamic loads on a starting configuration (zero deformation) under such conditions. These aerodynamic forces are then regarded as constant follower forces applied on the structure, and the correct amount of structural deformation, thrust force, flap deflection and gravity are computed to exactly balance this aerodynamic force and keep the airframe flying level at the fixed velocity. The solver then iterates the aerodynamic force by computing it under this new deformed configuration and repeats the process until convergence. By picking different angles of attack through the method of bisection, the solver eventually finds the correct angle of attack (with the converged thrust and flap deflection) that requires a gravity of 1g for level flight.
The trim angle of attack (computed at the centre node of the airframe), thrust and flap deflection for varying centre pod payload is shown in Figure 29, with comparison against Su [7] and Patil [8]. The comparison is found to be very good although the current method uses a reduced modal description for aerodynamic forces, the associated approximations made in order to pose it into such a form have contributed to the differ- ences seen in the results. The structural deformations at the trim conditions of the six different payloads are shown in Figure 30.
7.2.3 Trim Stability Validation
A linearized eigenvalue analysis of the system around the trim configuration is performed for each payload of the rigid/flexible case (Figure 31). Again comparison is made for the phugoid eigenvalue between the current method and previous works by Su [7] and Patil [8].
Here it is seen that as well as increasing in frequency, the phugoid mode becomes unstable in the flexible case after 50% payload, while in the rigid airframe the phugoid mode is always stable. This highlights the contribution of structural flexibility to the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 Ba se A O A / D e g Current Su Patil 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 T h ru st p e r En g in e / N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6
Payload / Max Payload
F la p D e fl e ct io n / D e g
Figure 29: Angle of attack, engine thrust and flap deflection at trim condition for payload varying from 0 to 100%, compared against previous results by Su et al. [7] and Patil et al. [8]
dynamic stability of this airframe and in particular a decreased level of stability in this model is associated with an increased bending deformation on the wing, as shown in Figure 30. The phugoid frequency and damping for different payloads matches very well with Patil’s results (while Su’s frequency tends to be lower). However the damping of the phugoid mode lies closer to Su’s results for the low-payload cases. Given the difference between the two referenced results, the current result is deemed to be in good agreement with them. The flexible phugoid becomes unstable soon after the payload reaches 50% of maximum, again agreeing well with both Patil’s (51%) and Su’s (61%).