1.1. An Introduction to Rockets, Dynamics, and Air Brakes
The flight path of a rocket can be broken down into four stages. At first, a rocket will launch, after which it experiences a powered ascent. Following its ascent until motor (propellant) burnout, it will begin coasting to its apogee, after which its final stage is parachute-deployment for the recovery of the rocket [
1]. Endeavour is a rocketry team at The University of Edinburgh and the flight path planned for their Darwin I rocket,
Figure 1, is shown in
Figure 2. In
Figure 1, the nose cone (left-most component) holds the payload, followed by the upper fuselage with the drogue chute, the avionics bay and the airbrake located just below, followed by the main chute, motor, fins, and the aft cone. The origin of the coordinate system is located at the tip of the nose cone, with
x-axis along the length of the rocket and
y- &
z-axis normal to the fuselage. From
Figure 2, as Darwin I reaches 2500 m after its powered ascent and subsequent burnout, it will be travelling at 250 m/s and will require the deployment of an airbrake to increase drag so that it may reach its target apogee of 3048 m. A few parameters of importance to the design of Darwin I airbrake are summarised in
Table 1.
A stable rocket will be able to recover its nominal flight trajectory even when some disturbance acts upon it [
1]. It will correct its non-zero angle of attack (AoA) from the disturbance through a corrective moment. The distance between the centre of mass (
) and centre of pressure (
) of a rocket is calculated to determine its stability. Often, this distance is shown as a multiple of its body diameter, with units of body calibre. A rocket is deemed stable if its
is behind its
, and as a general rule of thumb, the distance between the two should be between 1 and 2 calibres. The three main forces acting on a rocket are thrust of the motor, gravity, and the aerodynamic forces. The thrust is aligned with the velocity of the rocket, and produces no moments. Gravity acts on the
of the rocket, and also produces no moments. Only the aerodynamic force produces normal forces and moments on the rocket body, thereby affecting its stability.
Vick [
2] conducted a series of experiments on speed brakes with Mach Numbers ranging from 0.2 to 1.3 in fully developed turbulent flows. He investigated the effect of brake deflection angle, aspect ratio of brakes, brake chord height relative to boundary layer, and Mach number on drag. His findings are relevant to the airbrake as the geometry of the speed brake is similar to that of the airbrake. For all aspect ratios, the coefficient of drag (
) is dependent on the ratio of brake chord to boundary layer height, with the dependence growing stronger with reducing deflection angle. For a speed brake with a deflection of 90
, Vick [
2] showed that
is independent of aspect ratio and that at a Mach Number of 0.678, the
of a speed brake is 1.15. The boundary layer thickness at Mach Number 0.678 is 0.00279 m, or 2.79 mm (taken at a location where the velocity is 95% of the freestream). This value can be used to verify CFD simulation results, assuming fully developed turbulent flow. Flow before and after the airbrake can be approximated to flows over a forward-facing step and a backward-facing step. A forward-facing step is when the surface increases in height with respect to the free stream, and decreases for a backward-facing step. Both flow scenarios are nonlinear and can become unsteady [
3]. The separation of flow due to the forward-facing step induces large vortex shedding and Kelvin–Helmholtz oscillation in the shear layer. In general, there are three flow regions: upstream, downstream, and the redevelopment regions. The upstream region is created due to the adverse pressure gradient before the step surface, and the downstream region is located over the leading edge of the step. Then, the flow reattaches to the step face in the redevelopment region. Zhu and Fu [
4] investigated the forward-facing step in laminar free stream conditions. The numerical simulation conducted has shown that the step face amplifies the transition of boundary layer from laminar to turbulent. Here, the turbulence intensity parameter (
) is calculated according to Equation (
1), where
is the turbulent kinetic energy and
u is the magnitude of the local velocity. Ref. [
5] studied the turbulent subsonic flow over a backward-facing step. Similar to the forward-facing step, the flow is divided into three regions, with strong three-dimensional disturbances in the re-circulation region immediately after the step face, reattachment and settling of flow further downstream in the reattachment and transition region, and far downstream where the flow is relaxed and can be approximated as a two-dimensional flow.
The current airbrake that has been designed for Darwin I is the ‘Pancake’, which utilises a servomotor and actuation mechanism to extend three drag generating surfaces beyond the circumference of the fuselage,
Figure 3. The Pancake airbrake is designed to be manufactured using a combination of aluminium, mild steel and high density polyethylene (HDPE). While aluminium and mild steel are mechanically proficient materials, they are also high density and as such, will add undesirable weight to Darwin I. The Pancake design only generates 32 N of drag, which is close to ten times lower than the 305 N requirement for this rocket; however, if fully actuated from start to apogee, it will enable the rocket to reach its target apogee. It also requires 1.13 Nm torque to fully actuate, and its drag-to-torque ratio is, therefore, 28.3. The aims of this paper are redesign the Darwin I airbrake to (a) decrease the mass and volume of material whilst maintaining mechanical integrity used by using combinations of advanced composite materials and (b) increase the drag-to-torque ratio to improve the efficiency of drag generation by engineering lightweight composite folding mechanical metamaterials designed using principles inspired by the art of origami.
1.2. Origami Structures
Origami is an ancient art with deep-set Shinto and Buddhist roots. Movement in origami structures is enabled by spherical mechanisms around the vertices in fold intersections [
6]. An origami pattern is composed of three components; faces (or panels), fold lines, and vertices that define the fold lines. Each fold is assigned a mountain or valley fold [
7] with its own dihedral angle and fold angle. If a fold angle is positive (counter-clockwise rotation with respect to coordinate system), then it is a valley fold, and a mountain fold when it rotates clockwise [
8]. In recent times, the diverse configurations of origami have been considered for use in innovative mechanical metamaterials with application potential within a plethora of engineering sectors [
9,
10,
11,
12,
13]. The Miura-ori folding pattern has been of particular interest amongst researchers. Silverberg et al. [
9], for example, used this pattern to create a mechanical metamaterial structure with a negative (in-plane) Poisson’s ratio and a positive Poisson’s ratio (with the same magnitude as the negative Poisson’s ratio) in bending. The engineering applications for a system like this are vast including, e.g., impact absorption, deployable structures, and more. Similar principles of structural design can be applied to other origami patterns to control mechanical properties. Jiang et al. [
10] for example, designed octagonal building blocks with a Young’s modulus and Poisson’s ratio that could be fine-tuned as a function of the type of fold. Another Miura-ori application reported by Kshad and Naguib [
11] was capable of self-deployment and retraction via a shape memory polymer. Boatti et al. [
12] also used the Miura-ori fold but, in their work, to tune the thermal expansion behaviour, with possible applications in aerospace, optics, energy, and in microelectronics. Wang et al. [
13] demonstrated that both electromagnetic behaviour and chirality, could be fine-tuned using Miura-ori folds.
The fold (i.e., hinge) is a vital component for actuation. In paper, a material with negligible thickness, the folds are a ideal line segments that act as a rotational hinge [
8]. However, since a material with sufficient stiffness is required for the airbrake, it is most likely that the thickness of the panels will be non-negligible as thickness directly relates to stiffness [
14]. Origami structures with non-zero thickness material are called rigid origami. Thickness-accommodating models adapt different techniques to allow the kinematics of rigid models to be as close to the zero-thickness (paper) model as possible [
15]. Lang [
14] conducted a review of various thickness-accommodating techniques and the manufacturability of them. The techniques mentioned are: tapered panel technique (embedded zero-thickness model), offset panel technique, hinge shift technique, membrane technique, doubled hinge technique, rolling contact technique, and strained joint technique [
14]. Of particular interest to our work is the membrane technique as it can recreate the ideal hinges of the zero-thickness model through materials such as fabric. The membrane becomes a hinge with zero-thickness and can preserve the full range of motion of an origami pattern while achieving coplanar flatness. The minimum width of the membrane depends on the angle of the fold, with additional width depending on the required rigidity of the panel [
14,
15]. The limitations of this technique arise from the flexibility of the membrane material, which requires an understanding of the strain energy within the membrane for an accurate kinematic model. Further, numerical models for origami structures with membrane joints are useful tools in origami engineering and design [
16].
1.3. Actuation of Origami Structures
Deployable origami structures can be divided into passive and active origami [
8]. Passive origami is when the structure is deployed with a mechanical input (e.g., a servo), while active origami is actuated through non-mechanical fields/stimuli (electromagnetic, thermal, chemical). An airbrake designed through active origami could be actuated through shape transformations using, e.g., shape-memory alloys/polymers (SMA/SMP) activated by thermal stimuli, or using elastomers activated by changes in the electromagnetic field. Actuation with chemical fields would be difficult as it requires that the structure be submerged in liquid. SMAs are a group of alloys that returns to their ‘original shapes’ after thermally activated mechanical deformation. The temperature change required for this recovery depends on the alloy composition, ranging from 1 °C to 100 °C [
17]. Wood [
18] considered the application of SMA in a load-lifting and as supporting structures. SMA wires used in their tests were heated sequentially to lift a load, after which it would lock into a final configuration. The wires were placed strategically into folds with unique angles for each fold-configuration, enabling efficient actuation [
18].
With passive origami, a gear train or a rotating mechanism with rods similar to a piston cylinder (as has been used in the Pancake airbrake described above) can be used. An actuation mechanism studied by Faist [
19] is the Hoberman mechanism. It utilises rotation of a central ring to deploy the leaves of the solar panels attached to a CubeSat. The Hoberman mechanism allows the system to have one degree-of-freedom and, therefore, simplifies actuation. The Flasher is an origami structure that can be actuated and held in tension with a perimeter truss [
20]. In this paper, the truss was driven by wires and motors, while torsion springs located at the joints enable effective deployment. Other methods mentioned by Zirbel [
20] include pneumatic systems, centripetal acceleration with torsion springs, stored strain energy, and application of shape-memory polymer either as the main actuation mechanism or in view of aiding motion.
In this work, the origami Flasher was deemed a good origami candidate structure for an airbrake, as when closed (stowed) it is essentially a 3D structure that aligns flush with the fuselage of the rocket, while in its open state it becomes a flat 2D plate that extends beyond the diameter of the fuselage, enabling increased drag. We research these as composite structures attached to a perimeter truss that actuates passively.