Aeroelasticity is the study of the interactions between dynamic, inertial and aerodynamic forces that arise when a body is immersed in airflow. The unique challenge of aeroelasticity is to analyse how vibrations, static deflections and lift and drag forces combine, and to make sure that any interaction of these three forces does not lead to inferior aircraft performance or even failure.

The triangle in the figure below is known as Collar’s triangle and each vertex shows one of the forces mentioned above. When all three forces interact simultaneously we are in the realm of aeroelasticity and common failure modes include wing flutter and buffeting. When inertial and elastic forces combine in the absence of aerodynamic forces we are in the classical domain of structural dynamics and essentially dealing with any sort of mechanical vibration that you would experience on any piece of moving machinery.  The interaction of inertial forces and aerodynamic forces gives rise to aerodynamic stability problems. How does an aircraft react to small disturbances – do the oscillations dampen out or do they get worse over time? Finally, the interaction of aerodynamic forces and elastic forces can give rise to a phenomenon known as divergence, which is an effect where twisting of the wing becomes theoretically infinite and can cause wings to twist off.

The Collar Triangle defining aeroelasticity as “the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design.”

The two most dramatic aeroelastic effects are flutter and divergence. Flutter is a dynamic instability, often of the wing, caused by positive feedback between the wing’s deflection and the aerodynamic lift and drag forces. The flutter speed is the airspeed at which the wing, or any other part of the structure, starts to undergo simple harmonic motion – much like the simple to and fro motion of a simple pendulum – and this vibration occurs with zero net damping. Zero net damping means that there is no dissipation of energy (think of a pendulum swinging for eternity) and so any further decrease in net damping  will result in self-oscillation – the structure is basically forcing itself to vibrate more and more, which at some point, will naturally lead to failure.

As we all know, the lift force acting on a wing will tend to bend it upwards, but what is less well-known is that this lift force can also cause the wing to twist. This is because the centre of pressure, the point where the total sum of the lift pressure field is assumed to act on an airfoil, is not necessarily coincident with the shear centre, the point through which a bending load needs to be applied to get pure bending without any twisting. Imagine holding a ruler in one hand and pushing up on it with your other hand. If you apply the load along the central axis of the ruler, the ruler will only bend, but if you apply the load at one of the two sides you can see the ruler bend and twist ever so slightly. Most of the time, the shear centre of an airfoil is not coincident with the centre of pressure, and so a lift force produces both bending and twisting. A critical phenomenon called divergence can occur when this twisting of a wing increases the angle of attack, which consequently increases the lift force further or creates further mismatch between shear centre and centre of pressure, so that a feedback loop ensues until the wing diverges or essentially shears off. In fact, one of the Wright Brothers’ main rivals in the race to being the first at heavier-than-air flight was Samuel Langley, whose prototype plane crashed into the Potomac river in Washington D.C., and this is now believed to have occurred as a results of torsional divergence. Furthermore, torsional divergence was a large problem with many WWI fighter planes and required a lot of additional stiffening of the wings.

Divergence of wings in action

Forward-swept wings

One of the domains where divergence is particularly pernicious is in forward-swept wings. Simply put, wing sweep delays the onset of shock waves over the wings and therefore reduces the associated rise in aerodynamic drag caused by boundary layer separation. In slightly more detail, as air flows over a curved object, such as an aircraft wing, it accelerates due to centripetal forces and this means that an aircraft travelling slightly slower than Mach 1.0 (the speed of sound) can develop pockets of supersonic flow over areas with high local curvature, typically the wings or the canopy. For thermodynamic reasons, supersonic flows terminate in a shock wave which results in a sudden increase in the density of the air. This effect disturbs the smooth flow over the wing and creates vortices behind the aircraft, which means it is a form of parasitic drag. Sweeping the wing reduces the curvature of the body as seen from the airflow by the cosine of the angle of sweep. For example, a 45 degree sweep reduces the effective curvature by around 70% ($\cos 45^\circ = 0.71$) compared to the straight-wing case. As a result, this increases the airspeed at which supersonic pockets start to form by about 30%, such that the aircraft can reach speeds much closer to Mach 1 before shocks occur.

Another way to think about the effect of sweep is to imagine the airflow over the wing as shown in the figure below. The effect of sweeping is such as to break the airflow into a component normal to the wing chord (“normal component”), and one along the span of the wing (“spanwise component”). The maximum curvature of the wing occurs along the wing chord, and the normal velocity component for the swept wing ($V \cos \psi$) is always less than the normal component for a straight wing ($V$).

The figure above highlights another critical aspect of swept wings: the spanwise component. On a backward-swept wing the spanwise flow is outwards and towards the tip, while on a forward-swept wing it is inwards towards the root (see the figure below). Firstly, with the air flowing inwards towards the fuselage, wingtip vortices and the accompanying drag are reduced. Wingtip vortices form when the higher pressure air underneath the wing is sucked up onto the lower pressure top surface of the wing, thereby reducing the effective lift-generating surface of the wing. On most modern backward-swept airliners, winglets and sharklets prevent this phenomenon from occurring. Forward-swept wings similarly minimise this effect by re-routing some of the flow towards the wing root, and therefore allow for a smaller wing at the same lift performance. The second advantage of forward-swept wings is that shockwaves tend to develop first at the root of the wing, rather than towards the tips, and this helps to reduce tip stall. Aerodynamic control surfaces such as ailerons are typically located near the tips of the wings, because the further outboard, the greater their effect on controlling the rolling action of the plane. Tip stall essentially renders these ailerons useless, and therefore jeopardises the pilot’s control over the aircraft. As a result, the dangerous tip stall condition of a backward-swept design becomes a safer and more controllable root stall on a forward-swept design, providing better manoeuvrability at high angles of attack.

For all their merits, forward-swept wings suffer from one detrimental flaw – divergence. In a forward-swept wing configuration, the aerodynamic lift causes a twisting force that rotates the leading edge upward, causing a higher angle of attack, which in turn increases lift, and twists the wing further. With conventional metallic construction, additional torsional stiffening is typically required which adds weight, and is therefore sub-optimal in terms of aircraft performance.

Enter the Grumman X-29

The Grumman X-29 was an experimental aircraft developed by Grumman in the 1980’s, and flown by NASA and the US Air Force. The X-29 tested a forward-swept wing, canard control surfaces, and computerised fly-by-wire control to counter balance the various aerodynamic instabilities created by its airframe. From my perspective, the most important innovation, however, was the novel use of composite materials to control the aeroelastic divergence of forward-swept wings. At the time, composite materials were popular in the high-performance aircraft community as a means of creating stiff and strong structures at very low weight. In fact, composites were mainly used to save weight. However, the X-29 showcased a second advantage of this new material over classic metallic structures – multi-functionality.

Metals are isotropic materials, meaning that their stiffness is the same in all directions. The relationship between stress and strain along one direction of an aluminium panel is the same as in any other direction. Because composite materials are a union of stiff fibres held together by a resin matrix, we can manufacture panels that are stiffer in one direction than in another. This is because the composite material will be very stiff along the fibre direction but relatively compliant perpendicular to the fibre direction. In most fibre-reinforced composite materials, such as fibreglass and carbon fibre, this variation in stiffness is restricted to the plane of a single sheet of material known as an orthotropic lamina.

Consider one such layer of a continuous fibre-reinforced composite in the figure above. The material axes 1-2 denote the stiffer fibre in the 1-direction and the weaker resin in the 2-direction. If we align the fibres with the global x-axis and apply a load in the x-direction, the layer will stretch along the fibres and compress in the resin direction (or vice versa). However, if the fibres are aligned at an angle to the x-direction (of say 45°), and a load is applied in the x-direction, then the layer will not only stretch in the x-direction and compress in the y-direction but also shear. This is because the layer will stretch less in the fibre direction than in the resin direction. This effect can be precluded if the number of +45° layers is balanced by an equal amount of -45° layers stacked on top of each other to form a laminate, e.g. a [45,-45,-45,45] laminate. However, this [45,-45,-45,45] laminate will exhibit bend-twist coupling because the 45° layers are placed further away from the mid plane than the the -45° layers. The bending stiffness of a layer is a factor of the layer-thickness cubed plus the distance from the axis of bending (here the mid-plane) squared. Thus, even if the +45° and -45° layers have the same thickness, the outer 45° layers contribute more to the bending stiffness of the [45,-45,-45,45] laminate than the -45° layers do. Therefore, stretching-shearing coupling is eliminated in a [45,-45,-45,45] laminate as the number of +45° and -45° layers is the same, but bend-twist coupling will occur because the +45° layers are further from the mid-plane than the -45° layers.

Let’s now apply this effect at a wing level, i.e. a $[+\theta,-\theta]$ layup is used for the top wing surface and a $[-\theta,+\theta]$ layup for the bottom wing surface. At the global wing level, the layup  is balanced because we have an equal number of $+\theta$ and $-\theta$ layers, but the $+\theta$ layers are further away from the wing mid-plane than the $-\theta$ layers. This means that the bending stiffness is dominated by the $+\theta$ layers, and the wing will twist when it bends.

In the Grumman X-29, this bend-twist coupling was successfully exploited to prevent divergence in the forward-swept wings. As aerodynamic lift forces the wing tips to bend upward, the forward-swept wing wants to twist to higher angles of attack, but the inherent bend-twist coupling of the composite laminates forces the wing to twist in the opposite direction and thereby counters an increase in the angle of attack – divergence is avoided!

Bend-twist coupling in Grumman X-29 wings. Both top and bottom wing skin may have the same number of +theta and -theta fibre angles, but if the +theta angles are further from the wing mid-plane then they will dominate the bending behaviour and cause the leading edge to twist down as the wing bends up.

The Grumman X-29 is an excellent example of an efficient, autonomous and passively activated control system. Rather than adding more material to the wing to make it stiffer (but also heavier) an alternative solution is to use the bend-twist coupling capability of composite laminates. This capability is an example of elastic tailoring, and remains one of the most under-exploited advantages of composite materials. As the big aircraft manufacturers overcome the initial hurdles of using composites on a large scale with the 787 Dreamliner and A350-XWB, expect more and more of these multi-functional capabilities of composites to find their way onto aircraft components.