21st International Conference on Adaptive Structures and Technologies (ICAST) University Park, Pennsylvania, October 4–6, 2010

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Morphing Trailing Edges with Shape Memory Alloy Rods

Silvestro Barbarino*, Wulf G. Dettmer and Michael I. Friswell

School of Engineering Swansea University

Singleton Park, Swansea SA2 8PP, UK

ABSTRACT

Multiple flight regimes during typical aircraft missions mean that a single unique optimized configuration, that maximizes aerodynamic efficiency and maneuverability, cannot be defined. Discrete components such as ailerons and flaps provide some adaptability, although they are far from optimal. Wing morphing can significantly improve the performance of future aircrafts, by adapting the wing shape to the specific flight regime requirements, but also represents a challenging problem: the structure has to be stiff to maintain its shape under loads, and yet be flexible to deform without collapse. One solution is to adopt structural elements made of smart materials; Shape Memory Alloys (SMAs) have demonstrated their suitability for many static applications due to their high structural integration potential and remarkable actuation capabilities. This paper proposes an innovative flap architecture for a variable camber trailing edge, whose reference geometry is based on a full scale wing for a regional transport aircraft. The compliant rib is based on a truss-like structure where some members are “active” rods made of SMA. These actuators are able to sustain the external loads while allowing controlled shape modification. Elastic elements are incorporated to provide pre-straining for the SMA and therefore allow cyclic actuation. The layout of the structure is obtained using a preliminary optimization process, incorporating practical constraints, by focusing on a basic truss-like element and its repetition and position within the overall truss. The aero-structural performance is estimated using FE analysis. The SMA behavior is modeled using a dedicated routine to evaluate the internal stress state and the minimum activation temperature. The design fulfils a number of key requirements: a smooth actuation along the chord; morphed shapes that assure the optimal aerodynamic load distribution for high lift; light weight; low mechanical complexity. The design features of the architecture are investigated, and the requirements of the morphing skin discussed. Keywords: Morphing, Variable camber, Hingeless flap, Shape Memory Alloys, Genetic optimization 1. INTRODUCTION The geometry of a fixed wing is usually the result of a compromise design and its optimal performance occurs only in some flight regimes of a typical mission profile, typically for the cruise condition. The adoption of aerodynamic control surfaces (such as slats, flaps, etc.) allows the expansion of the domain of allowable flight regimes for an aircraft, but at the expense of poor performance or low aerodynamic efficiency. Thus, the idea of adaptive aerodynamic surfaces [1], that are able to change some features to fit the best optimal shape for a specific regime, promises interesting developments. Following this trend, many investigations have addressed components such as flaps, slats, by pointing out related benefits and drawbacks. More challenging strategies (e.g. variable sweep wings) have been taken into account, remaining, at least for the moment, confined to military applications (e.g. Northrop-Grumman F14).

Classical geometry-variation devices, like flaps, exhibit some advantages: they are easy to be integrated within the surrounding structure and do not compromise the wing’s structural integrity. Moreover, they affect a limited zone of the wing; this simplifies the actuation system architecture that can be practically concentrated close to the hinge zone. However, due to their concentrated nature, they produce discontinuities, sharpening of the geometry and worsening the aerodynamic efficiency. Moreover, connecting zones have to support the transmitted loads with inevitable structural weight increase. * Research Officer and corresponding author (s.barbarino@swansea.ac.uk, Tel.: +44 (0)1792 513177)

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To avoid these problems, novel strategies are being considered. For example, the idea of producing smooth variations of the geometry (i.e. Morphing) [2,3], even in presence of large displacements distributed over a wider portion of the wing, is well documented. The present study follows this trend, investigating the performance and the aerodynamic benefits which can be achieved by comparing a traditional split flap with a smooth, hingeless trailing edge. Based on the aerodynamic procedure discussed in more detail in paragraph 3.2, and assuming a hinge position at 0.7c, the wing airfoil portion between 0.7c and 1c has been rotated around the hinge point in order to simulate the split-flap down deflection (Fig. 1): assuming all other quantities are constant, the deflection angle δ has been considered in the range 2 - 24 deg to estimate the lift coefficient. In correspondence to a flap rotation δi, around the hinge point A, the trailing edge moves from the point P to the point P’. Then, for each hinged flap rotation δi, a corresponding morphed flap shape is found. In this case, the motion of point P to point P’ is obtained by morphing the flap camber line according to a parabolic arc connecting the point A to the point P’ and tangent in A to the unflapped wing airfoil camber-line (Fig. 1).

In Fig. 2, the wing airfoil Cl for a conventional split flap and a morphed flap are compared for each δi. For a given trailing edge down-deflection, the morphed flap solution leads to higher wing airfoil Cl than that for a conventional split solution.

Figure 1. Wing airfoil boundary: a) no flap rotation; b) split flap deflection; c) flap morphing.

Figure 2. Wing airfoil lift coefficient comparison.

This approach, however, gives rise to an interesting paradox: the same structure that has to endure the external aerodynamic loads without suffering appreciable deformations, has to dramatically change its shape to fit the best current flight condition. For this reason, several international researchers are working on this topic by following different approaches to significantly modify specific wing parameters (e.g., local or global camber [4-7], wing span [8-10] or twist [11], etc.) to achieve better aerodynamic performance. Some efforts have been focused on the development of kinematical chains, such as the Quaternary-Binary Cross Linked Mechanism (QBCLM) [12], others on the development of compliant structures [13]. Among these, the adoption of a Variable Geometry Truss (VGT) concept as primary load carrying structure is quite widespread [14-16]: a VGT is a truss structural system having a number of length-adjustable members.

In spite of the large effort, few flying prototypes have been produced. One noteable exception regards the work performed at NextGen Aeronautics [17], where a small scale aircraft able to double its wing span and surface area was designed and tested in-flight: in this case, the structure morphed by changing its planform through kinematic chains [18].

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Elsewhere, some experiences have been performed on the air inlets of fighter aircraft [19], adaptive chevrons aimed at further reducing jet noise [20], or even other aerospace fields, such as helicopters [21-27] or space applications [18,28,29].

One possibility to overcome the “morphing contradiction” is represented by the adoption of SMART materials (e.g. PZTs, magneto-rheological fluids, etc.), which gained momentum among scientists in recent decades by being able to play the two-fold role of structural and actuator elements. An example of these materials is Shape Memory Alloys (SMAs) [30,31], whose ability to recover large deformations upon thermal activation (Shape Memory Effect) while providing high transmitted forces, makes them a promising tool for morphing applications. Several applications adopting SMART materials in general, and SMAs in particular, for aircraft morphing can be found in the literature [18,32-36]: however, most research is still at the concept or early experimental stage, or has been used on small scale vehicles.

This work is aimed at controlling the airfoil camber at the wing trailing edge by producing a smooth morphed flap, substituting a traditional split flap on a full scale wing of a civil regional transportation aircraft. A 2D study has been developed to design a compliant rib. The idea is to combine the benefits of VGTs with some “active” rods made of SMAs, to numerically design an internal truss structure able both to withstand external aerodynamic loads and capable of changing its shape. Few investigations have been performed to date on variable geometry trusses with one or more elements made of Shape Memory Alloys [37,38], as proposed in this work, especially if used not only as actuators but also as structural, load carrying elements.

The compliance and change in geometry of the truss structure is significantly influenced by the SMA elements: their number and position within the structure, length and stress state. The SMA thermo-mechanical behaviour has been estimated by means of Liang & Rogers’ model [39], being able to estimate individually the minimum necessary activation temperature and the effective recoverable strain for each SMA element within the truss.

A Genetic Algorithm has been adopted to optimize the truss topology; its performance is evaluated in terms of achievable lift coefficient (which depend on the maximum displacement and shape of the morphing trailing edge), and required actuation temperatures for the SMAs. Some constraints have been imposed to the optimization process to attain reasonable truss structures: the main goal is to obtain a simple truss, suitable for future manufacturing and experimental prototyping.

Within the optimization algorithm, the design and the simulation of the architecture take advantage of a combined aero-thermo-structural numerical approach: a commercial FE code (MSC Nastran) is exploited to predict structural behaviour under the joint action of external loads and the SMA elements, evaluated with specific Matlab routines.

A preliminary discussion also addresses some feasible solutions for a morphing skin, capable of withstanding high deformations under aerodynamic loads with a smooth surface and without being affected by failure, to be used with this truss structure.

2. TRUSS DESIGN

A truss structure suitable for the purpose of shape adaptation should enable smooth shape changes, but retain a high degree of stiffness in other deformation components to allow precise operation by a manageable activation system [14]. For this purpose, Variable Geometry Trusses (VGTs) have been largely investigated: a VGT is a truss structural system having a number of length-adjustable actuated members [40]. Although the idea of substituting some of the truss elements with “active” rods is nothing new, only few studies have been carried out to date adopting Shape Memory Alloy rods within VGTs, both as structural and actuation elements. The introduction of a number of active elements in place of passive truss rods allow the structure to deform and thus increase the elastic energy. Moreover, conventional actuated length-adjustable truss members made of prismatic actuators based on a ball-screw mechanism or those based on hydraulic or pneumatic telescopic cylinders are of relatively high cost and heavy weight: from this viewpoint, the adoption of SMA rods as “active” members can help to overcome these limitations and allow for aeronautical applications of VGTs.

The truss structure herein developed is similar to a conventional mechanism, which behaves as a system of rigid bodies (members) interconnected by ideal joints: the study of such a mechanism can be performed using a kinematics approach. All of the truss rods, both passive and active, are supposed to be hinged at both ends (hence no bending actions are allowed): this assumption is particularly beneficial in the adoption of the Liang and Rogers’ model [39] for Shape Memory Alloys, which only accounts for axial stresses/strains. Moreover, at this design stage, SMA elements can be considered both as wires or rods, as they are always pre-strained and subject only to tensile actions: as discussed more in detail later, this is necessary to allow for cyclic actuation of SMA elements.

The basic truss element taken into consideration is a triangular truss in which one member is realized in “active” Shape Memory Alloy (Fig. 3); the additional elastic element, required to suitably pre-stress the SMA, will not be considered further here. A recoverable strain of 3% in length for the SMA elements has been assumed throughout this paper, as a compromise between actuation performance and stability with cycles [41,42].

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Assuming the left hinge is constrained, the kinematic investigation focused on how to improve the geometry variation of this basic triangle, upon SMA activation, by changing the angular aperture of the top vertex and the position of the connection point of the SMA on the rigid rod on the right. The range of top semi-angle is 5 - 85 deg, while the SMA connection point can move from 0 to 85% of the right rod length, starting from the bottom; the length of the rigid rods has been assumed equal to one unit (Fig. 4). The performance is estimated in terms of the total displacement of the free hinge on the right rigid rod (Fig. 5), as a percentage of the unit length of the rods.

Figure 3. Basic truss element.

a) b) c)

Figure 4. Superimposition of initial (black) and actuated (blue) triangle configurations: a) top angle is 20 deg and SMA position 0%, b) top angle is 100 deg and SMA position 50%, c) top angle is 160 deg and SMA position is 80%.

Figure 5. Contour plot of the right rod free hinge total displacement as a percentage of the unit length of the rods: blue to green to red represents increasing performance.

Some configurations, with small top angles and the SMA close to the top vertex, had to be discarded due to unfeasible configurations: the SMA activation in these cases can lead to an undetermined state, due to the superimposition of the truss elements and very high forces. Considerations arising from Fig. 5 are:

- as expected, the SMA length must be maximized to increase the geometry variation; - positions of the connection point between the SMA and the right rod which are close to the top vertex increase

performance, discarding the ones that lead to unrealizable configurations; - configurations with a big aperture at the top vertex perform better, but also lead to larger trusses.

Hinge

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In particular, the last consideration implies that the optimal configuration defined within the adopted reference geometry is a trade-off between the number of truss elements which can be fitted in the volume and their performance. It is not evident, as shown in the next paragraphs, that increasing the number of trusses leads to an improvement in performance. On the other hand, having fewer and wider trusses can not only increase overall geometry variation, but can also lead to lighter configurations.

Two main limitations exist for this basic triangular truss. Firstly, the SMA length affects performance but it is limited to stay within the volume of the single triangle: some solutions can overcome this restriction and are left for further developments, as discussed in the conclusions. Second, the basic triangle is capable of changing its geometry only in one direction, making it functional for a morphing flap sub-structure. However, the adoption of two triangles, one on top of the other (in a mirrored way), would allow for a two-way actuation, making this truss structure more useful for a wider range of applications. Such a solution has been considered for the morphing trailing edge: a sketch of a preliminary morphing rib composed of five trusses is shown in Fig. 6, also comprising of a stiff sliding panel solution, as discussed next.

a) b)

Figure 6. Preliminary morphing rib composed of five trusses and a stiff sliding panel solution (red lines represent activated SMA rods): a) downward and b) upward configuration.

3. OPTIMIZATION PROCESS

Design approaches for compliant mechanisms from the literature mostly involve the homogenization method [43], the ground structure approach or graph representation, adopted from conventional topology optimization. Based on these parameterizations, various approaches using different objective functions and constraints have been developed by several authors. Frecker et al. [44] and Kota et al. [45] adopted multi-criteria formulations by maximizing the mutual potential energy (to guarantee a proper degree of flexibility) and minimizing the strain energy (in order to achieve the desired structural stiffness). Sigmund [46] maximized the mechanical advantage, as a ratio between the output and the input force, by using spring elements at the mechanism’s output (this approach has been adopted by several authors) [47,48]. Another method was presented by Saggere and Kota [49], in which the force of a single actuator is transferred to a set of so-called active points on a flexible surface through the optimized compliant truss. Lu and Kota [50] formulated a general synthesis procedure for shape-adapting compliant mechanisms, using genetic algorithms to solve the design problem based on a binary ground structure approach. Another approach has been developed by Sauter [51], who used a graph-based optimization method involving evolutionary algorithms for the design of compliant mechanisms. To sum up, the presented optimization approaches have proved their ability to design compliant mechanisms with distributed compliance for one defined load case.

In this work, an optimization process based on Genetic Algorithms [52] has been adopted to find the best VGT topology, for the design of a compliant rib for controlling the airfoil camber at the wing trailing edge by producing a smoothed morphed flap, upon activation of the SMA elements.

A full scale typical regional transportation aircraft has been considered for the reference geometry, constraining the study to a 2D airfoil near the wing root with a chord length of 2.57m. In particular, the investigated morphing portion extends from 0.7c (c is the airfoil chord) to the trailing edge (0.77m) of the considered airfoil; a span strip (or rib spacing) of 0.3m has been assumed to estimate aerodynamic loads.

The optimization is focused on the design of the internal VGT, and the skin has not been taken into account at this stage: for this reason, the aerodynamic pressure loads have been concentrated on the external nodes (the ones laying on the upper and lower surfaces) of the truss structure. Some discussion on skin issues is given at the end of this paper.

Within the optimization algorithm, the design and the simulation of the morphing rib takes advantage of a combined aero-thermo-structural numerical approach. The Matlab code implementing the parallelized genetic algorithm is capable of integrating the contribution from different routines, each focused on a particular aspect of the optimization problem. After

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the generation of each individual within the optimization process, according to the genetic methodology, a preliminary check is executed on the newly created individual to verify its capability to withstand aerodynamic loads in the non-morphed configuration (SMA elements are passive at this stage): this allows the elimination of unsuitable solutions at the very beginning of the optimization iteration, reducing computational time. The aerodynamic pressure distribution is estimated by means of a 2D Vortex Lattice Method [53] on the input airfoil, suitably modified in the morphing portion according to the actual position of the external truss nodes. Then, aerodynamic loads are evaluated, and concentrated on the same external nodes of the truss for the Finite Element simulation, carried out using the MSC/Nastran solver, to get the mechanical and kinematic behavior of the morphing trailing edge. A suitable routine has been written to write Nastran input models, execute them and read the desired results of each simulation.

An additional test is also performed on the actuated model to eliminate impracticable solutions, which produce compenetration among elements, based on the intermeshing of segments (each representative of a rod) in the 2D plane: this validation is only executed between two successive trusses, considering all of the possible combinations.

Upon successful validation of the generated geometry, the optimization process takes place, as shown in the “iterative core” of Fig. 7. In this iterative process, the SMA behaviour is estimated for each SMA element present within the truss by means of an additional Matlab routine integrating the Liang and Rogers’ model: according to the stress level within the selected element upon thermal activation and due to external loads, the effective recoverable strain is calculated (and updated in the FE analysis), together with the minimum necessary activation temperature (useful for energy considerations). The iterative process terminates upon verification of the convergence check.

Figure 7. Conceptual scheme of the overall optimization procedure for a single individual of the Genetic Algorithm.

Additional details of these routines are described in the following paragraphs. Finally, a fitness function for each individual is estimated, according to the formula expressed in (1):

WT

Cfitness

MAX

l 1

1.0

102

(1)

where Cl is the lift coefficient generated by the airfoil integrating the current morphing trailing edge solution, TMAX is the max value of activation temperature estimated among all of the “active” SMA rods (the one belonging to the most loaded element) and W is representative of a preliminary weight estimation of the truss rib, based on the assumed cross section for each rod (kept constant, not optimized) and the calculated length of each member of the current truss. The first term represents an estimate of the achieveable performance vs. required actuation energy, while the second one takes into account structural efficiency. Note that some coefficients are introduced to bring both terms of the fitness function to the same order of magnitude, plus a weighting factor of two for the first term to enhance the focus on performance.

3.1. Genetic Algorithm

Genetic algorithms belong to the larger class of evolutionary algorithms, which generate solutions to optimization problems using techniques inspired by natural evolution, such as inheritance, mutation, selection, and crossover [52]. Each

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invididual (or configuration to be estimated) is characterized by its own “genetic code” or cromosome, generated as a set of values selected within suitable intervals for each optimization parameter.

Several parameters have been considered in the optimization procedure for this study: - the number of trusses to be fitted within the trailing edge volume which, in turn, defines the total number of nodes

and SMA elements; - the chord-wise length of the rigid portion at the trailing edge within the morphing volume, in which no truss

elements are present (this constrains the truss within a portion of the morphing volume close to the flap root); - the position of each node of the truss within the morphing volume, considering the ones constrained to lay on the

upper/lower surface and the internal ones laying on the mid camber line (a check to avoid possible superimpositions has been integrated too);

- the position, along selected “passive” rods of the truss, of one of the two extremes of each SMA element. Some restrictions are applied to the position of the internal nodes of the truss structure: in particular, these are

constrained to lay on the mid camber line, and each of them must be positioned within the trapezium defined by the four nodes of each truss laying on the upper and lower surface. This avoids the selection of very irregular truss shapes, which may not be suitable to carry external loads.

Regarding the selection of the SMA elements within the truss structure to be activated, these are selected at the beginning of the process and are not subject to optimization in this study. A “random” selection of active SMA elements made by the genetic algorithm would lead to an increased number of possible configurations, most of which would be unsuitable or not useful to the optimization target. In this work, interest has been focused towards a morphing flap or aileron, leading to the selection of the active SMA elements grouped together according to their relative position (upper or lower) to the camber line. Finally, it has to be noted that the estimated performance is actually the maximum performance, as all SMA elements on the same “side” of the camber line are considered active: several combinations with partial actuation are possible, leading to multi-stable intermediate configurations of the morphing trailing edge.

The genetic algorithm executes the iterative core, performing the simulation of the VGT structure actuated by Shape Memory alloys and in the presence of aerodynamic loads. The estimated quantities are the lift and pitching moment coefficients generated by the morphing airfoil, the structure nodal displacements and overall shape (defined by the external nodes on the upper and lower surface), and the required SMA activation temperature (as a measure of the required energy). The fitness function, from which the genetic algorithm determines how to optimize the final design of the morphing rib, is based on these quantities and has already been discussed.

3.2. Aerodynamic Estimate

A typical regional aircraft wing airfoil has been assumed as the baseline for the morphing hingeless application here presented (Fig. 8), focusing on the aft part of the airfoil extending between 0.7c and the trailing edge (the morphed flap portion has chord cf = 770 mm).

Figure 8. Wing airfoil geometry.

The aerodynamic loads computation assumes inviscid flow. The pressure distribution of a two-dimensional airfoil has been calculated using a Vortex Lattice Method: this 2D panel method includes the effects of the airfoil thickness, angle of attack, and mean camber line in the calculation of the aerodynamic pressure distribution [53]. The aerodynamic forces acting on the skin are calculated from the coefficients of pressure distribution (Fig. 9). The dimensional pressure on the i-th skin element can be estimated using

ipipi qCCVp ,,2

2

1 (2)

where 2

2

1 Vq is the dynamic pressure, Cp,i is the local pressure coefficient, V∞ is the free-stream velocity, and ρ is the

air density. The aerodynamic load Fn,i on each local skin element is then given by

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ipizin CqSnF ,, (3)

where nz is the vertical load factor and Si is the local strip area (as the product between the length li of the element along the airfoil cross section and the width wi along the span of the local segment). Cp,i is assumed to be constant due to the limited span of the investigated flap portion (wi = 300 mm).

Figure 9. Pressure coefficient distribution for the baseline airfoil geometry.

The aerodynamic load distribution acting on the flap portion was evaluated according to (3) with a unitary load factor, 5 deg angle of attack and a dynamic pressure equal to 1500.00 N/mm2. The distributed aerodynamic load on each skin element is then transferred to the FE mesh, concentrated as several nodal loads acting on the truss structure.

The aerodynamic loads are evaluated several times within the optimization process: on the baseline airfoil, to determine the load carrying capability of the newly generated truss for each individual of the Genetic Algorithm and discard unsuitable geometries; on the morphed truss, to estimate the effective actuation of the morphing portion within the FE approach. At each iteration, the external nodes of the truss which define the overall shape of the trailing edge within the FE model are extracted and, by undergoing a suitable change in reference systems, are substituted within the baseline airfoil to define a new morphed shape. The process is iterated until the convergence criterion is satisfied, and the final lift and pitching moment coefficient for the investigated individual can be estimated: these values then become part of the evaluation of the fitness function.

3.3. FE Approach and SMA Integration

To predict the performance of the morphed rib, an FE approach has been adopted using the MSC/Nastran solver: integration with the optimization process in Matlab has been possible by developing a specific routine to write the input model, using the geometry generated by the genetic algorithm and aerodynamic loads generated from the VLM routine, execute the simulation and read the results from the Nastran file format. All of the truss elements, both the passive ones made of aluminium and the active ones made of SMA, are hinged at both ends to create the desired mechanism: due to the expected large displacements and strains within the SMAs, a nonlinear analysis has been performed. At this stage, the focus is the topological optimization of the truss structure: for this reason, the cross section of the aluminium truss elements has been oversized (circular tube with a 10mm diameter and 1mm thickness) to avoid structural instabilities and is not considered in the optimization process.

SMA elements only work on tension (which allows the use of both wire and rod elements) as they are always pre-stressed to a suitable level necessary to induce the pre-strain and allow for cyclic actuation. A pre-strain value equal to 3% has been assumed in this study, as a compromise between actuation performance and stability with cycles [41,42]. From a practical perspective, this means that additional elastic elements or hinges with a torsional stiffness would be required within the truss structure to pre-strain the SMA elements: these elements also stiffen the structure to external loads and improve its dynamic behaviour. From a simulation point of view, the SMA elements in the FE analysis are assumed to be already pre-strained and can recover up to 3% of their initial length upon activation. However, the pre-stress level must be added to the actual stress state when evaluating the SMA effective behaviour (recoverable strain and activation

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temperature). For this reason, and since no SMA model is implemented within the Nastran FE code, the active elements are assumed extremely rigid and able of fully recovering the imposed negative expansion coefficient (SMA elements contract as the temperature grows): failure to do so would lead the Nastran solver only to partially recover the imposed strain. Direct integration of the SMA constitutive law has been successfully achieved by the first author [32] and many researchers [54-56] in several FE solvers, but there is little documentation of integration with the Nastran solver. An external Matlab routine has been written implementing the Liang and Rogers 1D model [39]: since all of the elements are pinned at both ends, only axial actions are allowed (no bending actions), and this model can be effectively used; moreover, in this phase the real cross section has been considered for the SMA rods (assumed as solid rods with a 5mm diameter), leading to the correct estimation of the structural behaviour of the alloy. Thus, a suitable procedure has been realized in order to simulate the action of the SMA elements, integrated within the rib structure. Passive SMA rods are simulated in the FE model by considering the properties of Table 1for the martensite phase.

Shape Memory Alloys are materials that exhibit the Shape Memory Effect, that is, regaining its undeformed original shape when heated, after deformation performed at a low temperature. An SMA has two crystalline phases called Martensite and Austenite. In the Martensite phase at a low temperature, the material yielding stress is low and a deformation can be applied with a relatively small applied force. Heating the SMA causes the phase transformation from the Martensite to the Austenite, recovering the deformation and returning to its initial shape: when constrained upon recovery, the SMA is capable of generating a larger force than that applied to pre-strain it. Heating can be performed externally or internally: in the latter case, the Joule effect can be used since SMA is a conductive material.

SMAs are ideal for their favourable actuation performance per weight ratio: one of the major current drawbacks in engineering applications is the difficulty of fully characterising their behaviour, which is highly non linear and load history dependent.

The Liang and Rogers’ model was developed in 1990, based on Tanaka’s phenomenological approach [57]. However, even adopting the same constitutive law, Liang and Rogers assumed a simpler cosine relationship for the phase kinetics to describe the martensite fraction as a function of stress and temperature:

2

cos2

MASA

M bATa Austenite Marteniste

(4)

2

1cos

2

1 AMFM

A bMTa

Marteniste Austenite

where SFA AAa / and FSM MMa / are combinations of material constants: AF, AS, MF, MS are material

characteristic temperatures, respectively known as austenite final, austenite start, martensite final and martensite start. They refer to phase transformations of the material and define when a particular crystal structure appears or the conversion is complete. Two other combinations of the material constants are

AAA Cab / and MMM Cab / , where CA and CM are the

slopes of the straight lines approximating the functions relating the four temperatures to the acting stress. Finally, ξM and ξA are the martensite and austenite volume fractions present in the alloy before heating and cooling, respectively. The necessary material constants have been experimentally computed and refer to a commercially available Ni-Ti-Cu alloy (Table 1).

Table 1. Thermo-mechanical properties of the adopted Ni-Ti-Cu Shape Memory Alloy.

Martensite Young modulus, EM [GPa] 16.5 Austenite Young modulus, EA [GPa] 40.2 Ultimate elongation [%] 12.0 Max (assumed) recoverable strain [%] 3.0 Min necessary pre-load [MPa] 155.6 Martensite start temperature, MS [°C] 43.3 Martensite finish temperature, MF [°C] 27.5 Austenite start temperature, MS [°C] 47.3 Austenite finish temperature, MF [°C] 58.9

In Fig. 10, the logical scheme adopted for simulations is illustrated. At first, an FE non-linear analysis on the structure is

performed for the maximum imposed value of pre-strain (3% in length) applied to each active SMA element by means of an equivalent thermal coefficient, in order to relate the forces to be provided by the SMA elements to the displacements transmitted to the structure. Then, the pre-stress level required to apply the 3% pre-strain at the next activation cycle is added and the corresponding curve is compared with the SMA force-displacement curves, estimated by the Liang and Rogers model for several temperature values. Among the different curves, the Matlab routine finds the one corresponding to

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the minimum activation temperature, whose intersection with the structure curve gives the effective recoverable strain. Finally, the equivalent thermal expansion coefficient for each SMA element is updated in the FE simulation to meet the solution found by the SMA model.

On convergence, the results from the FE simulation are available to the genetic algorithm in terms of nodal displacements of the truss elements (and so, the overall shape of the morphing rib) and stress levels within the SMA elements, together with the minimum required activation temperatures.

a) b)

Figure 10. SMA model integration within the FE approach.

4. NUMERICAL RESULTS

The geometry of the morphing truss structure is created by the genetic algorithm, for each individual, according to the parameters (optimization variables) and constraints (to achieve suitable designs) discussed in the previous paragraphs. The threshold for the validation and acceptance of the geometry at the preliminary check stage has been imposed as a vertical displacement at the trailing edge lower than 1% of the flap length (chord-wise): for the optimized solution here presented, this value is 0.55% (equivalent to 4.28 mm). The iterative process has a threshold which is based on the relative trailing edge displacement between two subsequent iterations, which has to be lower than 1 mm (0.13%).

Figure 11. Evolution of the genetic algorithm optimization process.

Only geometries generated by the genetic algorithm which satisfy the preliminary check, the intermeshing test between elements and reach convergence in the iterative process are then estimated in terms of the fitness function (otherwise

SMA recoverable strain and activation temperature

F.E. Model

(Thermal)

SMA Model

(Liang & Rogers)

SMA applied force (internal stress) + required pre-stress for cyclic actuation

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assumed equal to zero): for this reason, the optimization algorithm has been set up to generate large generations (400 individuals), to investigate a wide selection of different geometries and compensate for the large number of deleted individuals, especially at early stages of the process.

The considered fitness function has been discussed already in paragraph 3. Convergence of the genetic algorithm is reached after 75 generations (Fig. 11), showing no additional improvements up to 40000 investigated configurations (100 generations). The maximum value achieved for the fitness function is 12.168, corresponding to the optimized morphing truss shown in Fig. 12b: black lines represent passive rods made of aluminium, red ones refer to SMA elements (although not necessarily activated); cross sections are not represented.

The downward actuated configuration is show in Fig. 12c: a vertical displacement at the trailing edge of 237 mm (30.7%) is achieved in the presence of aerodynamic loads, producing estimated lift and pitching moment coefficients equal to 3.19 and -1.29 respectively (these may be over estimated due to the adopted VLM); maximum activation temperature for the most loaded SMA rod is estimated as 73.7°C. The upward actuated configuration is show in Fig. 12a: this is not part of the optimization process at the moment and is evaluated only to verify the two-way capability of the proposed concept. A vertical displacement at the trailing edge of 37 mm (4.7%) is achieved in the presence of aerodynamic loads, producing lift and pitching moment coefficients equal to 0.18 and 0.06 respectively; estimated weight of the rib with the assumed cross sections is 284g, while the maximum activation temperature is 68.1°C in this case.

Figure 12. Geometry of the optimized morphing truss generated by the genetic algorithm (red dotted lines represent SMA elements): a) upward actuated, b) undeformed and c) downward actuated.

Table 2. Required activation temperatures for the SMA elements.

SMA active element Temperature, °C

Downwards deflection

Upwards deflection

Truss 1 (root, left)

SMA upper 0 65.0 SMA lower 71.7 0

Truss 2 (middle)

SMA upper 0 64.6 SMA lower 73.7 0

Truss 3 (tip, right)

SMA upper 0 68.1 SMA lower 65.6 0

The detailed activation temperatures for the selected SMA material are reported in Table 2. These activation

temperatures depend on the maximum stress level to which each active element is subject upon activation in presence of

a)

b)

c)

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aerodynamic loads. Once the SMA material (a good selection of the material composition can lower the required energy) and the cross section of the adopted active elements is selected, it is still possible to lower the required activation temperature for some of the active elements in the truss by considering two SMA elements in parallel instead of one. Thus, the cross sectional area is doubled and the stress levels can be divided, which also lowers the energy which must be supplied (if, for instance, Joule effect is used for heating); however, this produces an increase in total weight of the solution. A trade-off design can be then achieved and further optimization can be performed.

Some considerations can be made on the morphing rib solution found by the genetic algorithm. The optimization process selected as better performing those structures having a small number of trusses (actually 3). This shows that the influence of the top angle of each triangle on performance is greater than the number of trusses within the same volume. Moreover, this structure has fewer elements and therefore is less complex and lighter. The overall shape of the morphing truss, which can be estimated by connecting the external nodes laying on the upper and lower surface, is quite smooth during downward actuation.

However, the two-way capability of such a solution is very limited, as the upward motion does not produce remarkable performance nor a smooth shape (the morphing capability is limited to the airfoil tip): this behavior could be expected, as the fitness function adopted in this work does not take into account these effects, and further improvements are required.

For instance, proceding with a manual search among the solutions generated by the optimization process, it is possible to find another individual whose performance is very close to the “optimal” solution (fitness of 11.853), but with improved behavior in the two-way actuation (see Fig. 13).

Figure 13. Geometry of a sub-optimal morphing truss (red dotted lines represent SMA elements): a) upward actuated, b) undeformed and c) downward actuated.

Table 3. Required activation temperatures for the SMA elements.

SMA active element Temperature, °C

Downwards deflection

Upwards deflection

Truss 1 (root, left)

SMA upper 0 66.7 SMA lower 71.6 0

Truss 2 (middle)

SMA upper 0 65.7 SMA lower 73.6 0

Truss 3 (tip, right)

SMA upper 0 74.8 SMA lower 67.6 0

a)

b)

c)

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In this case, the downward actuated configuration (shown in Fig. 13c) has a vertical displacement at the trailing edge of 233 mm (30.2%), producing estimated lift and pitching moment coefficients equal to 3.05 and -1.23 respectively (lower than the optimal case). The upward actuated configuration (Fig. 13a) has a vertical displacement at the trailing edge of 85 mm (11.0%), producing lift and pitching moment coefficients equal to -0.37 and 0.31 respectively (better than the optimal case). Table 3 summarizes the activation temperatures estimated for this solution, while weight has been calculated as 280g.

5. SKIN ISSUES

Although many morphing aircraft concepts have been elaborated, only a few deal with the problems relating to a smooth and continuous skin that simultaneously deforms and carries loads. A morphing skin can be envisaged as an aerodynamic fairing to cover an underlying morphing structure. A requirement for use in an aerospace application is that it must be able to change shape in at least one of two ways: either changes in surface area (e.g. flaps, slats on an aircraft wing) or changes in camber (e.g. aileron, flaps, slats, winglet on an aircraft wing or variable pitch propeller). This shape change can be instigated either by external or integrated actuators which would make the skin self-actuating or active and potentially “smart”. To meet desirable shape changes, stiffnesses can either be tailored or actively controlled to guarantee flexibility in the chord-wise (or span-wise) direction with tailored actuation forces. Hence, corrugated structures, segmented structures, reinforced elastomers or flexible matrix composite tubes embedded in a low modulus membrane are all possible structures for morphing skins.

Thill et al. gave an extensive review of the state of the art for morphing skins [61]: according to the authors the importance of a compliant and stiff morphing skin has been recognised but not many satisfactory solutions have been proposed. While most of the axial and bending loads are counteracted by the stringers, ribs and spars, the skin needs to withstand a combination of tensile, compression and shear forces (semi-monocoque structures, i.e. a stressed skin concept) and needs to be able to distribute the diffuse aerodynamic pressure loads to the underlying structure. This highlights the difficulties inherent in replacing the current stiff and strong solutions with a more flexible approach.

In this work, aerodynamic loads have been concentrated on the external nodes of the morphing truss to check its load carrying capabilities within the optimization process; however, the skin surface is responsible for transferring the pressure loads to the underlying structure.

A preliminary solution investigated by the authors is the adoption of a stiff panel solution (i.e. made of typical aeronautical aluminum), which is constrained at the trailing edge and capable of sliding inside the wing box at the flap root upon shape change. Such a simulation has been implemented in the Nastran FE solver by means of the SLIDE LINE element which “connects” the nodes of the skin panel with the external nodes of the truss, allowing the sliding effect to take place. For the sub-optimal morphing truss shown in Fig. 13, the adoption of such a sliding skin produces the effects visible in Fig. 14 (rods cross sections are represented): in particular, while the optimized downward deflection produces the expected smooth shape, the opposite actuation generate an anomalous aerodynamic shape, calling for further improvements.

The sliding boundary condition must be suitably designed so as not to interfere with the desired smooth aerodynamic shape, especially on the upper surface. The sliding condition on the lower surface is less critical.

This skin solution can be traditionally stiffened in the span-wise direction with stringers, being careful to avoid overlapping with the truss rib. Additional elastic elements can be suitably designed and positioned at the flap root to help the sliding effect while keeping a smooth shape under aerodynamic loads (which could induce abnormal bumps on the surface). Moreover, these elements further stiffen the structure (this additional contribution should also be taken into account when estimating the total required actuation energy) and can help reducing unwanted dynamic phenomena (for instance, flutter).

a) b)

Figure 14. Preliminary morphing truss rib with stiff sliding panel solution: a) downward and b) upward configuration (in blue are represented rigid rods, in black passive SMAs and in red active SMAs).

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The adoption of a sandwich skin for the morphing truss rib is another viable solution: due to the higher bending stiffness of sandwich panels, the overall structure could be simplified (i.e., not needing the additional elastic elements at the flap root) while still guaranteeing a smooth external shape and adequate stiffness. To achieve the morphing capabilities, mainly 1D in the chord-wise direction with camber variation, such a sandwich panel could be realized by coupling an accordion-like honeycomb core with elastomeric faces, based on similar work proposed by the first author [27].

The topology of the proposed morphing rib could also allow the skin to be realized as alternating stiff, non-morphing panels (mainly attached or hinged to the external nodes of the truss) and stretchable panels (in between two external nodes). This last solution could also allow increased torsional stiffness between two parallel ribs by means of the stiff skin parts, which should then be suitably sized to account for the lack of a completely stiff skin.

Finally, “irregular” morphing truss structures, such as the one suggested by the optimization process (Fig. 12 and Fig. 13), require different levels of local strain on the skin surface in between two subsequent external nodes of the truss, according to their relative distance and the amount of required shape change. This observation leads to the conclusion that a morphing skin solution must be tailored to the host morphing truss structure. Therefore, the truss structure, together with the external loads, defines the requirements for the final skin solution, chosen as the optimum among the different possibilities. Solutions realized as a combination of multiple, different types of skins (traditional stiff panels coupled with reinforced elastomers, etc.) are also possible, adapted to the local morphing and loads carrying needs.

6. CONCLUSIONS

This paper has investigated an innovative flap architecture for a variable camber trailing edge, whose reference geometry is based on a full scale wing for a regional transport aircraft. The compliant rib is based on a truss-like structure where some members are “active” rods made of SMA, able to sustain the external loads while allowing controlled shape modification.

The layout of the structure is obtained using a preliminary optimization process based on a genetic algorithm, incorporating practical constraints, by focusing on a basic truss-like element and its repetition and position within the overall truss. The aero-structural performance is estimated by coupling a VLM in Matlab with a Nastran FE analysis; the SMA thermo-mechanical behavior is modeled using a dedicated routine to evaluate the internal stress state and the minimum activation temperature. The design features of the architecture have been investigated, and the requirements of the morphing skin discussed.

A fitness function incorporating the lift coefficient, maximum SMA activation temperature and estimated rib weight has been used to find the optimal solution. A rib truss characterized by low mechanical complexity and smooth morphing shape change has been found by the optimization process, able to achieve a lift coefficient larger than three.

As stated in the paper, the optimization process of the morphing trailing edge would benefit from also considering within the fitness function the two way actuation and an additional check on the smoothness of the morphed shape. Additionally, the actual check on compenetration between two rods of the truss would also benefit from considering not only intermeshing (based on the mean line), but also interference due to the effective cross section of each element.

Another improvement which can be implemented in the optimization algorithm is related to weight estimation: before considering its value in the fitness function, it is possible to adjust individually the cross section of each rigid rod according to its stress level (both for the axial and bending stiffness), following a pre-determined design criteria. With such an additional optimization, it becomes important to keep track of the structural weight within the objective function of the optimization process.

In this work, the estimated performance is actually the maximum performance, as all SMA elements on the same “side” of the camber line are considered active. Several combinations with partial actuation are possible, leading to multi-stable configurations of the morphing trailing edge. Another degree of freedom which could be left to the optimization algorithm is the selection of the active SMA elements: the genetic process could select which elements to activate, instead of manually pre-selecting them, leading to a broader investigation of different solutions, suitable for several applications (flap, aileron, spoiler, etc.). Regarding the activation temperatures, it has already been stated that they are related to the selected Shape Memory Alloy and the equivalent cross section of each “active” rod, which can be suitably changed by modifying the number and section of SMA elements for each truss. After having chosen the best trade-off solution for each active rod, a more effective estimation of the required energy can be calculated and considered within the fitness function.

Thus, a full set of parameters is available to find the best performing solution from multiple perspectives: ‐ Performance, by means of lift and pitching moment coefficient, nodal displacements and overall shape; ‐ Energy, by means of the required activation temperature and selected heating method of each active SMA element; ‐ Weight, by means of adaptation of the cross section of each truss element to the stress level it has to withstand.

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As an additional remark, it can be noted that due to the large variety of design parameters available, it becomes difficult just to define a single suitable fitness function, able to fully describe the optimization problem in all its complexity: further developments will be considered in this direction.

Further studies will also focus on techniques to increase the overall performance of the morphing truss, for instance extending the connection point of the SMA with the rod right to the outside of the triangle, by means of an additional rigid member perpendicular to the rod itself. Another way to achieve similar benefits is to attach the ends of each SMA rod to rigid rods of two different trusses, including non consecutive ones. Both solutions provide longer SMA rods and, thus, larger actuation capabilities; however, they require additional checking for intermeshing among the truss elements.

As a future development, the final truss solution, resulting from the optimization process, can be transformed into a compliant mechanism by substituting the internal hinges with “solid-state” hinges [58], bringing it within the classical category of mechanisms with lumped compliance [59]. Such solid-state hinges can be realized as short-length regions with reduced cross-section and, therefore, low bending stiffness. This allows the conventional mechanism to be transformed easily into a compliant one, only affecting the traditional hinges, but the capacity to carry external loads can be reduced. Moreover, mechanisms with lumped compliance typically show discontinuities in their deformation pattern, which makes them unsuitable for the purpose of shape adaptation [60], where a continuous deformation field is required.

An analysis of the dynamic behaviour of the structure also has to be addressed.

ACKNOWLEDGEMENT

The authors acknowledge funding from the European Research Council through grant number 247045 entitled “Optimisation of Multiscale Structures with Applications to Morphing Aircraft”.

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