Shape morphing is defined as changing the configuration of a wing in order to control the aerodynamic characteristics so that the wing may comply with the aircraft’s mission profile. The morphing wing has been investigated by several groups such as the Active Aeroelastic Wing Program of NASA and the Morphing Aircraft Structures Program (MAS) provided by Defense Advanced Research Projects Agency (DARPA). Smart materials have also been researched in conjunction with the morphing wing investigations.
According to the research conducted by Rodriguez (2007), four approaches for morphing wing technology may improve aircraft performance: expand the wing’s flight envelope, replace conventional control surfaces for flight control for performance and stealth advancement, reduce drag to improve range and reduce vibration, or control flutter. The benefits and future concepts of a fully morphing aircraft configuration have been investigated (Moorhouse et al., 2006). Furthermore, several concepts that focus on morphing skins were studied, such as corrugated structures, segmented structures, reinforced elastomers or flexible matrix composite tubes embedded in a low modulus membrane (Thill et al., 2008). In addition, wind tunnel tests were performed testing the morphing wing. Unsteady pressure signals were recorded and visualized in real time while the wing was being deformed to reproduce various airfoil shapes with controlling the two actuators displacements (Popov et al., 2010a, b).
In Sofla et al. (2010), morphing wing concepts were classified into three groups: planform alternation, airfoil adjustment and out-of-plane transformation. With planform alternation, wing area manipulation techniques are employed through which the wing’s span and chord length are resized. Alternatively, airfoil adjustment entails designing and changing the wing profile. Out-of-plane transformation consists of altering the chordwise and spanwise camber. The folding wing is capable of adjusting the flight performance from a cruise configuration to a high speed dash configuration. During the intermediate changing configuration, aerodynamic characteristics varied dramatically and quickly.
Finite element modeling, aerodynamic loading, and a continuous-time multi-body simulation were used to simulate the aeroelasticity of the wing of an unmanned combat aerial vehicle (Scarlett et al., 2006; Snyder et al., 2009). A methodology for the model and an analytical solution were developed in order to produce flutter solutions for a continuous representation of a two-segment uniform folding wing (Liska and Dowell, 2009). Moreover, a flutter analysis simulation was developed and tested while accounting for the nonlinearity of a wing structure (Attar et al., 2010; Tang and Dowell, 2008).
The unsteady vortex lattice method (UVLM) was developed to predict the geometry of the wake and the aerodynamic loads on the wing configuration in an incompressible flow. UVLM provides a highly effective solution of a complex problem with reasonable computational efficiency. The side-edge suction force for isolated or interaction planform was computed by applying the method (Lamar et al., 1975). Also it was applied to aircraft engine designs to improve the overall performance of both military and civil aircraft. UVLM was coupled with a typical section cascade structural model to form a set of aeroelastic equations of motion (Thake, 2009).
Most studies concerning the folding wing were performed under the assumption that the fold angle remained constant. However, this assumption produces varying results for a wing that is in motion. Accounting for this motion is a necessary action when designing the folding wing structure. In this paper, an aerodynamic analysis of a wing in the motion was performed by using UVLM.
Figure 1 shows the schematic configuration of a wing at an instant of folding motion.
In the figure, both φ_{1} and φ_{2} may be simultaneously varied during flight for efficient flight performance. These parameters affect aerodynamic characteristics such as lift and drag. The relationships between the global coordinates and local coordinates of each wing component are expressed as
where L_{1}, L_{2} are the span length of the inboard and the outboard component, respectively, and Λ is the sweptback angle.
The fluid surrounding the body was assumed to be inviscid, irrotational, and incompressible over the entire flowfield, excluding the solid boundaries and wakes of the body. According to Katz and Plotkin (2001), a velocity potential Ф(X, Y, Z) can be defined in the inertial frame; the continuity equation in this frame becomes
and the first boundary condition requiring zero normal velocity across the solid boundaries is
where v is the velocity of the surface and n is the vector normal to the moving surfaces. Also the second boundary condition is diminishing the flow disturbance far from the body.
where R=(X, Y, Z).
For unsteady flow, the Kelvin conditions supply an additional equation in determining the streamwise strength of the vorticity shed into the wake. In general, the condition states that in the potential flow region the angular momentum cannot change, and thus the circulation Г around a fluid curve enclosing the wing and its wake is conserved:
From the continuity equation, the general solution can be obtained by integrating the contribution of the basic solution of source σ and doublet μ distributions over the surface and wakes of the body,
Although, this formulation does not directly include a vortex distribution, doublet distributions can be replaced by equivalent vortex distributions.
On the other hand, the wake shed from the trailing edge of the lifting surfaces can be modeled by vortex distributions. The strength of each vortex Г_{Wi} is equal to the vorticity shed during the corresponding time step Δt such that
Consequently, the strength and location of the vortex must be specified for each vortex element. The wake vortex location should be aligned with the trailing edge and be located closer to the latest position of the trailing edge.
The zero normal velocity boundary condition on the surface (Q_{nk}=0) for an arbitrary collocation point k can be obtained the following equations.
Once the computations of the influence coefficients and the right-hand side vector are completed, the zero normal flow boundary condition on all collocation points of the wing will result as
And then the circulation can be obtained from this equation. Furthermore, the aerodynamic forces can be computed by using the Bernoulli equation and the pressure difference can be as follows.
where Δc_{i, j} and Δb_{i, j} are the panel lengths in the i-th and j-th directions, respectively. Similarly, τ_{i} and τ_{j} are the panel tangential vectors in i and j directions.
Then the load of this panel is
Therefore, the total forces and moments are obtained by the load of each panel. (Editor’s Note: This section is very nicely written)
Fold angle and angular velocity were the most important parameters in the numerical simulation. These parameters potentially change the aerodynamic characteristics of a wing while the wing is in motion. The simulation and verification were performed by C++ and MATLAB.
The wing consisted of three parts: the inboard and outboard components which undergo rotating motion and the body component that is not subject to motion. For this simulation, only the inboard component was assumed to rotate about the x-axis from 0° to 85° ø_{2} was set to zero.
The location of the wake behind the trailing edge was varied according to fold angle. Thus, if the wing is in motion,the wake will follow the trailing edge, and then the effects of the wake will not be same for a constant angle. The location of the wake in motion is shown in Fig. 2.
Additionally, the collocation points of each aerodynamic panel were changed during fold motion and then the circulation in Eq. (9) will be changed. As a result, the varied circulation value of the panels will produce differing aerodynamic characteristics. Therefore, the characteristics
[Table 1.] Model configuration (unit: inch)
Model configuration (unit: inch)
of the wing were varied for the wing in fold motion.
Table 1 represents the model details (Lee and Chen, 2006). This model is an innovative aircraft design proposed by Lockheed Martin as part of the DARPA MAS (Lee and Chen, 2006).
For the verification of this work, an aerodynamic analysis was performed using a simple uncambered, rectangular wing model that was suddenly set into a constant-speed forward flight, as given by Katz and Plotkin (2001). The results were almost equal to the data of the reference as shown in Fig. 3.
The freestream velocity was 50 m/s, the air density was assumed as 1 kg/m^{3} and the angle of attack was 5°. Also, in order to reduce the effect of sudden acceleration, the folding was initiated at 2 seconds. As shown in Fig. 4, the total lift
coefficient and pitching moment coefficient decreased.
If the aerodynamic forces orthogonally acting on the individual panels were taken into consideration, the tendency of the transient lift coefficient could be slightly different from that given in Fig. 4. But, the z-direction forces were only considered in the analysis, which was the direction of the total lift.
In Fig. 5, the sectional lift coefficient is shown. The coefficient of the inboard component largely decreased.
The distribution of the sectional lift changed as the fold angle reached 90°. As shown in Fig. 5, aerodynamic characteristics of the wing in fold motion varied. Additionally, the decrease of the lift coefficient of the inboard component was observed as the fold angle increased.
The pressure coefficient vs. nondimensional chord-wise direction of each part is represented in Fig. 6.
Large variations did not occur except for in the inboard component, similar as Fig. 5.
The fold angular velocity,ø was another parameter taken into consideration. The large fold angular velocity produced
opposite effects in the aerodynamic characteristics in comparison to the small angular velocity.
In Fig. 7, the variations of transient lift coefficient, drag coefficient and pitching moment coefficient are represented for each angular velocity.
From these results, a larger angular velocity gained more lift, drag and also pitching moment. Moreover, the lift coefficient decreased and the drag coefficient increased as the fold angle increases except case of 120 rad/sec.
An aerodynamic model for a folding wing was developed and vortex lattice method was used for the analysis. The model was derived from the DARPA MAS and analyzed for various fold angles and fold angular velocities.
The location of the wake behind the trailing edge varied according to the fold angle and the fold angular velocity. Thus, aerodynamic characteristics such as lift, drag and moment of the wing in fold motion also varied during flights.Also as the fold angle increased, the lift coefficient and the drag coefficient decreased. This indicates that the folded wing configuration was more suitable to dash. However, the results of this paper showed that the effect of a large fold angular velocity was also an important factor that cannot be ignored.
According to the results obtained in this study, the various aerodynamic characteristics in fold motion affect aeroelastic characteristics. Therefore, flutter analysis of a folding wing during the motion will be investigated.