Special Issue: AFOSR
Aero-structural optimization and actuation
analysis of a morphing wing section with
embedded selectively stiff bistable elements
Journal of Composite Materials
2023, Vol. 57(4) 737–757
© The Author(s) 2023
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DOI: 10.1177/00219983231155163
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José R Rivas-Padilla1 , D Matthew Boston2, Karthik Boddapati1 and Andres F Arrieta1
Abstract
Morphing wings provide a potential avenue to improve aerodynamic performance of aircraft operating at multiple
design conditions. Nevertheless, morphing wing design is constrained by the mutually exclusive goals of high loadcarrying capacity, low weight, and sufficient aerodynamic control authority via conformal shape adaptation. This tradeoff can be addressed by exploiting the stiffness selectivity and shape “lock-in” properties enabled by using bistable
beam-like elements within compliant structures. In this paper, we present an aero-structural optimization method to
realize morphing structures with selective stiffness and shape “lock-in” capability from embedded bistable elements.
We leverage an embeddable beam element with an invertible curved arch that provides stiffness selectivity and camber
variation to the proposed rib geometry. Optimization objectives and constraints are designed to maximize the
structure’s stiffness change and camber morphing “lock-in” effect when operating at two distinct flight conditions.
Using the optimization results, we manufacture a wing section demonstrator with selective stiffness and “lock-in”
morphing featuring two optimized ribs, a load-carrying skin made of a carbon reinforced laminate, Macro-Fiber
Composite (MFC) actuators, and a servo-controlled mechanism for switching the bistable elements’ states. The power
and energy requirements of actuating and holding a target deflection are experimentally measured and compared. The
results show that the bistable elements can assist in holding a target deflection at a reduced energy cost. Finally, we test
the experimental demonstrator in a low-speed wind tunnel demonstrating the load carrying capability and lift variation
achieved from switching states.
Keywords
Bistability, compliant structures, morphing structures, nonlinear mechanics, optimization
Advances in adaptive structures and materials have led to
renewed interest in conformal morphing aircraft in the
aerospace community.1 These designs are advantageous
because they allow for optimal operation of an aircraft
subject to a multi-objective mission with unique requirements and operational conditions at each stage.2,3 Shapeadaptable solutions must be concurrently lightweight in
construction, compliant enough to allow for adequate
control surface authority, and simultaneously able to resist
aerodynamic loads with small shape deformation.4 These
constraints impose an inherent design trade-off in the realization of conformal morphing structures, which can be
referred to as the morphing structures trilemma.
Analyzing shape adaptable systems in terms of energy
and work, we can categorize morphing structures as active
or passive.5 Compliant structures capable of sustaining
external loads and embedded with an actuator system are
categorized as active-load bearing structures (hashed domain in Figure 1(a)). The work done by the actuators and
external loads must be balanced with the internal strain
energy of the system when designing active load-bearing
structures. A potential route to alter this balance is by exploiting elastic instabilities6–10 to design embeddable bistable elements.11,12 Bistable elements allow the designer to
store and release strain energy by switching between stable
1
School of Mechanical Engineering, Purdue University, West Lafayette, IN,
USA
2
School of Aeronautics and Astronautics, Purdue University, West
Lafayette, IN, USA
Corresponding author:
Andres F Arrieta, School of Mechanical Engineering, Purdue University,
177 S Russell St, West Lafayette, IN 47907-2050, USA.
Email: aarrieta@purdue.edu
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Journal of Composite Materials 57(4)
Figure 1. Active-load bearing structure design domain integrated with selectively stiff bistable elements: (a) Design domain represented
as an energy balance between external loads and actuator work, and the internal strain energy of a compliant structure (the
intersection of these sets represent the design domain of active-load bearing structures); (b) Expanded design domain by storing strain
energy via integrated bistable elements (the green annulus region highlights the expanded design domain and added functionality by
exploiting the structural non-linearities of bisable structures); (c) Representation of the expanded functionality in terms of increased
output displacement achieved with bistable and selective stiffness behavior; and (d) Distinct structural response by leveraging stiffness
selectivity (δx is the state switching induced displacement; and K1 and K2 are the stiffness variability response of the initial and deformed
states, respectively).
states.13,14 The switching of the bistable element expands
the design space of active load-bearing structures with no
modification to the amount of work done by external loads
or actuators. This design domain expansion for the activeload bearing structures is highlighted by the green shaded
annulus domain in Figure 1(b). For example, in a multistable soft robotic gripper15 it would be possible to carry
heavier objects, since an actuator consuming a fixed
amount of energy could first lock the gripper in place by
switching between stable states, and then apply additional
continuous actuation to increase its gripping force. This
concept has been demonstrated by coupling the stored
strain energy of a bistable soft robotic gripper with additional pneumatic actuation to achieve an increase in the
maximum weight and dimensions of the object held by the
gripper.16
The bistable elements presented in this work are geometrically bistable (GBS), implying that their bistability
requires no external loads to be obtained in a similar way to
bistable domes,17 and in contrast to classical post-buckled
beams18 or pre-stressed counterparts.19 The characteristics
of GBS elements allow them to be embedded within a larger
compliant structure and display two distinct structural responses about the undeformed (zero stress) and deformed
states. Inducing a state switch requires transverse loading,
allowing the GBS to support significant axial loads without
experiencing eversion20 or a jump to the other available
equilibrium states. The process of state switching results in a
length change (δx) and a considerable stiffness reduction
along the axis of the element (henceforth referred to as
stiffness selectivity). Figure 1(c) shows the concept of how
this stiffness reduction in combination with the state induced δx augments a desired displacement output given a
fixed amount of work exerted on a mechanism (the area of
each bar represents the amount of work required to reach
a target displacement). Another consequence of the
stiffness reduction (K1 → K2) is a decreased input load (P)
requirement after switching from State 1 to State 2
(Figure 1(d)).
There is considerable previous work on the optimization
and integration of bistable units to facilitate the design of
compliant morphing wings.21–25 These bistable structures
could be used to achieve different types of morphing,
i.e., span-wise26–28 and camber morphing.29,30 In this
work, we present an aero-structural optimization methodology for the design of morphing wing sections exhibiting selective stiffness and shape “lock-in” capability
from embedded bistable elements. We first introduce the
GBS element and quantify its stiffness selectivity and
shape “lock-in” behavior. We then propose a set of optimization objectives and topology generation methodology
for the design of a bistable morphing rib topology with
integrated actuators and a compliant corrugation. The internal shape of the rib is optimized to switch between stable
states and operate at two distinct flight conditions associated with the respective undeformed and camber
Rivas-Padilla et al.
morphed shapes. Macro Fiber Composite (MFC) actuators
are selected to control the trailing edge deflection about
each stable state. At the heart of this study is also the
objective of shedding light as to whether MFC actuators,
that rely on very small mechanical advantage due to the
thickness of the piezoelectric element, can efficiently introduce strain energy for achieving controlled deformations about each stable state. A numerical comparison
analysis on the effectiveness of using MFC actuators and
servo actuation strategies is presented.
We manufacture a single modified version of the
optimally obtained rib geometry to validate the predicted
structural behavior. This single rib is then used to construct a wing section comprised of two ribs and covered
by a composite laminate skin integrated with a double
walled square corrugation31 on the lower wing section
surface. Finally, we characterize the aeroelastic performance of the wing section in a low-speed wind tunnel test
at speeds between 15 m/s and 28 m/s. The obtained
experimental results suggest the potential for actuation
effort reduction, which is correlated to weight,32 when
operating a bistable compliant wing about a specific
stable state for a prolonged amount of time, and the wind
tunnel tests show promise for achieving large changes in
aerodynamic performance without triggering dynamic
instabilities.
Selective stiffness and shape lock-in effects
from geometrically bistable elements
The potential for exploiting stored strain energy in compliant systems requires consideration of two key effects as
a consequence of the nonlinear behavior of bistable elements. Namely, the selective stiffness capability and the
ability to “lock-in” on a target shape without the need of
external actuation to hold it. Selective stiffness can be
defined as the ability to choose between two or more
responses to an external load with the same structure. In
the context of this work, it refers to the ability to select
between two global stiffnesses by switching the state of the
embedded GBS element (Figure 2). Selective stiffness in
compliant structures has previously been proposed as an
alternative to bridge the gap between the trade-offs faced
when designing morphing systems.33,34 This type of
stiffness variability can be achieved by exploiting local
changes in compliant topologies. Several approaches to
achieve local bistability within compliant structures have
been explored in the literature, including leveraging
thermally prestressed composite laminates,19,35–37 employing thin shells,38,39 and utilizing bistable compliant
mechanisms.24,40 The advantage of the thin shell elements
is that their bistable properties depend primarily on their
geometry. This type of GBS element is more convenient to
739
embed into compliant structures, since it can be integrated
as a monolithic component of a larger structure through
additive manufacturing methods. In a past study,41 numerical results demonstrated the potential for aerodynamic
gains in terms of lift-to-drag ratio by embedding these
geometrically bistable elements into a compliant rib topology. In the same paper, the aerodynamic performance
of this new geometrically bistable rib was compared with
those previously reported in,42 where embeddable bistable
prestressed composite laminates were used instead. These
studies focused on a camber morphing flap-like concept
with selective stiffness behavior. In the case of the GBS
element, the change in stiffness is a consequence of the
post-buckled behavior of the embedded arch geometry,
which yields a more compliant response when the element
is loaded along its axis. The flexible state in the design of
these morphing rib topologies corresponds to the higher
energy state, camber morphed configuration. In practice,
this structural response is best suited to low speed loiter
condition or for a short distance take-off/landing maneuver. aerodynamic performance for these maneuvers.
The stress-free stiffer configuration would then be
switched on for a higher speed cruise condition, where less
deflection of the trailing edge is required to maintain lift.
This design approach was inspired by the results shown in
Ref.43, where the authors perform an optimization study to
reduce drag at two distinct flight conditions (Mach =
0.100 and Mach = 0.417). They conclude that a higher
camber is required to maintain a target lift value at the
lower speed while a more streamlined configuration is
ideal for the higher speed condition.
The nonlinear behavior of the embedded bistable beamlike element and the two main effects we exploit are shown
in Figure 2. The selective stiffness behavior of the bistable
element is coupled with a state induced displacement or
change in length δx. Notice that the possibility to obtain
stable deflected shapes without continuous provision of
actuation is a key characteristic enabled by state switching
in multistable structures. Here, the distance between two
points connected by a similar beam element could be
reduced after switching between states. In our case, adequate positioning of the GBS element would achieve a
state induced camber variation of an airfoil without the
need of constant actuation. The deformation induced by
the state switching is reversible (i.e., elastic) and no work
input is required to hold it. This effect can be used to
program into the morphing structure several statically
stable shapes to “lock-in” and operate optimally at multiple design conditions when performing various flight
maneuvers. We focus on leveraging GBS elements as an
alternative for realizing selective stiffness on this paper.
However, the methodology presented can be adapted to
different elements, providing adaptable compliance from
multistability.
740
Journal of Composite Materials 57(4)
Figure 2. Selective stiffness and shape “lock-in” capability of geometrically bistable elements embedded in a morphing rib topology.
Displacement in the out-of-plane direction (y-direction in Inset 1) stores energy in the element until it switches state. Each state
exhibits a unique response to an in-plane (x-direction in Inset 1) load. Inset 2 demonstrates the camber morphing achieved via GBS
element state switching.
Design of the geometrically
bistable element
The design of the GBS element is derived from a concept
trim tab utilizing pre-stressed composites.44 Judicious design of the central arch structure results in a topology with
two stable states: one stress-free, and one deformed
(Figure 3). The stable deformed state occurs due to out-ofplane loading causing the arch to buckle and evert its
curvature, resulting in the in-plane loading of the flexural
members on either side of the arch. This in-plane loading of
the flexural members resists the tendency of the deformed
arch to switch back into its unstressed state, allowing it to
maintain a second stable shape. In this paper, we use a
geometry similar to that explored in Ref. 45, with the
geometric parameters shown in Table 1 and labeled in
Figure 3(a).
A three-dimensional finite element (FE) shell model is
created to evaluate the structural response of the GBS element. The simulations are carried out using the Static/
General Implicit ABAQUS Finite Element Analysis software, with reduced integration, linear elastic shell (S4R)
elements. The transition between stable states is accomplished via arch eversion achieved from the applied out-ofplane displacement on the curved region (Figure 3(b)). The
critical out-of-plane load to switch between states was
23.1 N, which is reached after 6.1 mm of displacement. The
total amount of displacement applied was 20 mm. To
quantify the stiffness change, the model is displaced inplane along the negative x-direction about each stable
configuration, following a procedure from Ref. 41, Measuring the reaction force at the pinned support versus the in-
plane displacement of the sliding support (Figure 3(c)),
provides a value for the in-plane stiffness of each state. The
results show that a stiffness ratio larger than 120 is possible.
Additionally, the distance between the end points of the
element is reduced by δx = 8.6 mm when switched from
State 1 to State 2. The state induced change in length is
leveraged in the design of our morphing rib to “lock-in” on a
camber morphed configuration. It is important to note that
this stiffness difference and change in length is subject to a
specific length of the element (GBS Element Length =
130 mm). During the optimization, discussed in the next
section, this length is free to change. Although the results
are limited parameter-set specific, they illustrate the general
nonlinear structural response of the embeddable GBS
element.
Aero-structural optimization of the
bistable morphing rib geometry
Selective stiffness can provide a route for achieving optimal
behavior in different operational conditions. This requires
developing a design tool capable of leveraging the topology
and positioning of the elements providing the stiffness
selectivity. We illustrate this by considering a rib topology
designed to operate at two distinct flight conditions, namely,
loiter and cruise maneuvers at velocities of Mach = 0.04 and
Mach = 0.10 (V∞ = 15 m/s and 35 m/s, respectively), respectively. These relatively low-speed flight conditions
were selected based on current UAV design trends.46–48 The
rib profile selected is a NACA0014 airfoil with a chord of
400 mm. This profile and dimensions were selected to have
Rivas-Padilla et al.
741
Table 1. General geometric parameters of GBS element.
Parameter
Dimension
Units
Arch region length
Element width
Inflection Radius
Flexural member width
Curve Height
Shell thickness
Inclination angle
44.0
40.0
3.5
6.0
11
0.75
2.0
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[°]
Figure 4. Schematic showing definitions for internal structural
topology. The topology features eight movable structural nodes
to connect the truss topology (green), two nodes to determine
the corrugation location and length (yellow), and one discrete
variable to determine the location of the bistable element (red).
Figure 3. Thinly curved geometrically bistable (GBS) element: (a)
Top and side view with geometric parameters, (b) FE model
with boundary conditions and results for State 1 and State
2 configurations, and (c) In-plane (x-direction) stiffness numerical
response results for the GBS element.
sufficient space for the internal structural elements and to
ultimately design a wing section that could fit in Purdue
University’s Boeing Wind Tunnel, where aeroelastic tests
are conducted. Nevertheless, the developed tool can be
readily modified to fulfill different requirements, including
using more modern airfoil families.
The internal topology is parameterized using ten movable nodes (Figure 4). The x-positions of eight structural
nodes are determined as design variables by the optimization tool, and the y-position of each node is constrained to
a region in space: three nodes are constrained to the upper
surface of the profile; three are constrained to the lower
surface; and two internal nodes are constrained to the line of
symmetry of the airfoil. The eight structural nodes are
connected using a Delaunay Triangulation49 to generate the
internal truss elements of the morphing rib topology. The
corrugation location is defined by a forward and aft node
which are positioned along the rib’s lower surface. A reduced order homogenized model of a double-walled corrugation (see Appendix A) is used to allow the necessary
in-plane extension and compression of the lower rib surface.
The details of the FE analysis of the compliant rib are given
in Appendix B.
The last three design variables define the location of the
GBS element (via a discrete variable) and the required
actuation voltage of the MFCs in the stiff and flexible state,
which we use as a standard trailing edge deflection control
smart actuator in this initial part of the investigation. The
internal beam elements generated by the positioning of the
structural nodes and the Delaunay Triangulation define
the possible locations of the geometrically bistable element.
The discrete variable selects which of the generated truss
elements will be swapped by the GBS element. Specifically,
this variable determines which of the truss elements from
the internal topology element connectivity list is best suited
to become the selectively stiff GBS element. In this initial
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Journal of Composite Materials 57(4)
Figure 5. Topology optimization algorithm with a nested weakly coupled aeroelastic loop. The algorithm implements Matlab’s Genetic
Algorithm to handle the evolution of discrete and continuous variables through each generation.
topology generation methodology, each of the generated
truss elements is considered as part of the structure. A truss
removal scheme was not considered for simplicity.
The bistable element’s stress-free shape is selected for
the undeformed state of the morphing rib. In contrast, the
second stable state is designed to deflect the trailing edge
increasing the airfoil’s camber to maximize the CL/CD ratio
at the desired flight condition. In our past results we have
demonstrated that it is possible to reduce the stiffness of the
structure by switching between stable states 41. This stiffness selectivity is leveraged to reduce actuation level required at the low flight speed condition (V∞ = 15 m/s).
We employ MATLAB’s optimization toolbox genetic
algorithm (GA) to find the optimal value of the 13 design
variables described above. The genetic algorithm is a mixed
integer solver capable of handling the discrete nature of the
design variable that identifies the bistable element. This
algorithm does not guarantee that a global optimum has
been found. To mitigate this, we conducted several optimization runs in preliminary studies and determined that it
was suitable for identifying viable solutions that would
illustrate our concept. The optimization algorithm combines
ABAQUS CAE to generate the topology and is weakly
coupled with the XFOIL50 aerodynamics analysis tool to
evaluate the aeroelastic response of each individual. A
schematic of the optimization algorithm is observed in
Figure 5. The genetic algorithm evaluates a population of
150 individuals over a maximum of 80 generations to
identify the optimal topology. The objective function is
given below:
Minimize:
f ¼ α1 f1 þ α2 f2 þ α3 f3 þ Pf
(1)
g1 ¼ 1 Cl, flex 0:800
(2)
g2 ¼ 1 Cl, stiff 0:147
(3)
f1 ¼ Cl, flex Cd, flex
(4)
f2 ¼ Cl, flex Cl, stiff
(5)
f3 ¼ Vstiff þ Vflex
(6)
Subject to:
where,
are the objective functions maximizing lift-to-drag ratio ( f1 ),
maximizing the lift variation between the stiff State 1 and
flexible State 2 ( f2 ), and minimizing the voltage for the MFC
smart actuators ( f3). The step penalty function for unsatisfied
constraints is,
Pf ¼
2
X
i¼1
¼
100*gi
gi > 0
0
gi ≤ 0
(7)
Finally, we use the scaling coefficients:
α1 ¼ 1=110
(8)
α2 ¼ 2=11
(9)
Rivas-Padilla et al.
743
α3 ¼ 1=1500
(10)
to treat this multi-objective problem as a scalarized single
objective one for convenience.
The scaling factors for each objective function are
heuristically chosen so that each objective exhibits equal
importance and yield values between 0 and 1. Convergence
is achieved when the delta of the pseudo objective function
changes by less than ϵ = 0.001 for 5 consecutive generations
or when a maximum of 80 generations is reached.
The constraint functions gj are designed to satisfy the lift
requirements at each flight condition. Since the lift increases
quadratically with the velocity, a lower lift coefficient is
required at faster flight speeds. Conversely, a higher lift
coefficient is necessary to achieve an equivalent lift at
slower flight conditions. Therefore, a target Cl,flex = 0.8 is
required for the flight condition of V∞ = 15 m/s, while a
lower Cl,stiff = 0.147 is necessary to achieve the same target
lift force at the higher flight speed of V∞ = 35 m/s. From this
point forward, the second structurally stable state of the rib
will be associated with the lower flight speeds because it
exhibits a state induced trailing edge deflection, increasing
the airfoil camber to satisfy the higher Cl,flex constraint.
The actuation method, and direction of applied force, to
switch the rib between states was not considered as part of
the optimization, since establishing a constraint of precisely
how the state switching force should be applied could
limit the solver to a sub-optimal solution of the internal
shape of the rib. Consequently, this sequential (i.e., postoptimization) exploration of the actuation methods limits
the presented work, as it does not initially seek to obtain a
weight-optimized state switching actuator, but rather
characterize which type of actuation strategy is most efficient for switching states and controlling the trailing edge
deflection. A more robust optimization methodology to
evaluate the weight and actuation trade-offs is outside of this
work’s scope.
Optimization convergence results
The optimal layout of the truss members and location of the
bistable element are shown in Figure 6. The location of the
bistable element is adjacent to the corrugated skin to allow
for the expected camber morphing, and the obtained optimal
topology satisfies the lift constraints while maximizing the
lift ratio between the stiff (State 1) and flexible (State 2)
configurations. The solution yields a maximum 2D lift-todrag ratio of 99.87 in State 2, a lift coefficient of 0.1692 and
1.22, in State 1 and 2 respectively, as well as an optimized
actuation voltage of 239.82 V and 169.95 V, in State 1 and
State 2 respectively (Figures 6(a) and (b)). This solution was
typically reached by the optimizer after 21 generations.
Figure 6(c) shows the fitness function’s evolution for each
generation. This is the result of using an initial topology
Figure 6. Optimal topology result: (a) State 1 deflection with
MFC actuation overlaid against undeformed state, (b) State
2 deflection with MFC actuation overlaid against undeformed
State 1, and (c) Fitness function evolution through each
generation.
optimization to generate sub-optimal candidates for the first
generation. More information on this process is provided in
Appendix C.
2D Aeroelastic response of the optimized rib
The optimal rib topology is studied in detail using a
standalone instance of the 2D aeroelastic tool used for the
optimization. This was done to characterize the optimized
aerodynamic performance as the MFC actuation voltage is
swept from 0 V to 900 V while the rib is held at α = 0°. We
observe a considerable lift-to-drag ratio increase via
morphing just from the state induced camber change
achieved by storing strain energy via the geometrically
bistable element’s state switch (Figure 7(a)). However, the
lift-to-drag ratio quickly drops as the actuation voltage of
the MFCs increases beyond 90 V(Figure 7(b)). The results
also show a clear difference in the external actuation requirements between State 1 and State 2. The stored strain
energy of State 2, achieved via the switching of the bistable
element, allows for a greater lift coefficient increase with
little additional actuation. We also observe that the stiffer
State 1 achieves the required lift coefficient for the flight
speed of 35 m/s at actuation voltages larger than 200 V.
Finally, the aerodynamic response achieved via camber
morphing about each stable state exhibits higher lift-to-drag
744
Journal of Composite Materials 57(4)
Figure 7. Aeroelastic result: (a) Drag polar comparison between
State 1, State 2 and a solid NACA0014, (b) lift-to-drag ratio
comparison of State 1, State 2 and a solid NACA0014.
ratio values than a rigid NACA0014 airfoil that is pitched
from 10° to +10°.
Single morphing rib structural response
To validate the numerical model presented in the optimization, a physical specimen is fabricated based on the
optimized rib topology using a conventional and additive
manufacturing (AM) process. This single rib specimen has a
width of 40 mm and is modified to feature an actuation horn
structure that, when pulled forward along the chordwise
direction, is capable of everting the bistable element’s arch
into the second stable state (Figure 8(a)). The topology was
also fitted with an aft-slanted, double-walled corrugation to
accommodate the necessary in-plane extension and bending
of the morphing rib lower surface when morphed. The
change of stiffness corresponding to each state of the rib is
measured using an axial testing machine (ATM) and the
experimental results are used to validate the finite element
model and optimization procedure. The fixture head was
fitted with a load sensor that recorded the reaction force on
the fixture head to displace the trailing edge (Figures 8(c)
and (d)). Good agreement is observed between the numerical and experimental responses (Figure 8(b)), thus
validating the modelling approach used for the optimization. An approximated stiffness increase of 3.5 times is
observed when switching from State 2 to State 1.
Figure 8. Images of the single morphing rib test: (a) A control
horn is integrated to the GBS element design and a doublewalled corrugation is embedded into the lower rib skin surface.
(b) Results of the stiffness test demonstrating the state induced
deflection and stiffness selectivity. The experiments show good
agreement with the numerical model. (c) The experimental test
setup showing the undeformed rib in State 1. (d) The morphing rib
in State 2 with the trailing edge displaced by tensile testing
machine fixture.
Additionally, a state induced trailing edge deflection of
around 16 mm can be observed as a consequence of the
stored strain energy caused by the eversion of the arch in this
bistable system. The state induced deflection increases the
airfoil’s lift and lift-to-drag ratio without the need of additional actuation. The benefits in power consumption and
the economy in energy spent by everting the bistable
Rivas-Padilla et al.
element to achieve this state induced camber morphing are
discussed in more detail in the next section.
Numerical actuation study
A numerical comparison between actuation methods is
performed to evaluate: 1) The control authority of the MFC
acuators (Method 1); 2) the morphing rib deflection with a
servo-actuated rigid control rod (Method 2); and 3) the rib
actuation with an antagonistic force pair generated by
servo-driven nylon wires connected to evert the bistable
element from one state to another (Method 3). Figure 9
illustrates the three actuation methods and the induced
elastic strain energy on the structure by each strategy. This
approach is used to determine which method induced less
strain energy on the system when deflecting up to the
optimal point. The results show that implementing a
conventional actuation strategy (Method 2: path [1] → [3],
solid blue line) with a rigid control rod induces the least
amount of elastic strain energy upon the system, while the
MFC actuation strategy (Method 1: path [1] → [2], solid
black line) induces 20 times more energy to deflect the
trailing edge to its aerodynamically optimal point. In
between Method 1 and 2, lays morphing the rib by antagonistically pulling on the control horn of the bistable
element (Method 3: path [1] → [4] → [5], solid red line)
induces 7.5 times more strain energy than Method 2, but
after reaching the stable equilibrium point (point [6] in
Figure 9), no additional actuation to hold the deflection is
745
required. An interesting observation is that the numerical
results predict a snap-back effect (path [4] → [5]) when
switching from State 1 to State 2 via Method 3. Specifically, to reach the State 2 equilibrium point at around
18 mm of deflection, the trailing edge must travel up to
32 mm (path [1] → [4]), at which point the rib initiates the
switching to State 2, springing back to 18 mm of deflection
at near constant elastic strain energy (path [4] → [6]). To
switch back from State 2 to State 1, the rib follows a
different more direct path (path [6] → [1], solid cyan line)
to the zero deflection stress free state. This completes the
actuation cycle of Method 3. These results imply that the
MFCs may not be well suited to actuate this type of
morphing system. First, they do not allow for providing
enough mechanical leverage to morph the rib and switch
the state of the bistable element from the undeformed
(point [1]) to the optimal position (point [5]). Secondly, the
excess strain energy needed to deform the rib can be
thought as forcing the structure to deflect to the desired
position (optimal deflection) through a much more rigid
deformation mode, compared to Method 2 and 3. Therefore, coupling actuator motion with low energy deformation modes is crucial to design an effective active-load
bearing morphing structure. Rather than investigating this
last point, we focus on determining the power consumption of the considered approaches given that these metric is
strongly correlated to actuator weight (see Ref. 51 for a
detailed discussion). We conduct this evaluation in the next
section.
Figure 9. Numerical results comparing actuation methods. Method 1) A high voltage source is applied to the MFC actuators which
induce a moment on the upper surface and deflect the trailing edge downward. Method 2) A servo is connected with a rigid control rod
to the trailing edge and rotating the servo arm clockwise achieves the desired downward deflection. Method 3) The servo arm is
connected with nylon wires to the control horn structure, and rotating the servo produces an antagonistic force pair to deflect the
trailing edge downward (a counter-clockwise rotation deflects the trailing edge downward). If rotated enough, it serves the double
function of switching the bistable element from State 1 to State 2 (the element can be switched back from State 2 into State 1 with a
clockwise rotation).
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Journal of Composite Materials 57(4)
Demonstrator manufacturing and power
consumption assessment of
actuation methods
We fabricate a morphing section with two adjacent ribs and
conduct structural and wind tunnel tests to assess the efficacy of the considered actuation methods in the previous
section.
Manufacturing of demonstrator
The wing section is fabricated from two compliant ribs
additively manufactured with a Markforged X7 FDM
printer using a carbon micro-fiber reinforced nylon filament
(OnyxTM). A double-walled corrugation (described in
Appendix A) is designed and manufactured from thermoplastic polyurethane (TPU) in an Ultimaker S5 3D printer,
following the polymer fusing technique described in Ref. 52
The top and bottom skins consist of a single 3-ply layup
([0,0/90-pw,0]) carbon fiber reinforced polymer (CFRP)
laminate, using pre-impregnated plain weave (Toray T300/
Newport 301) and unidirectional fiber (Grafil TR50S/
Newport 301) fabrics. Because the plate is thin (approximately 386 µm), it can be wrapped around and bonded to the
pre-assembled, 3D printed frame, following a procedure
similar to that used in Ref. 53 Additionally, four 8557-S1
MFC actuators are bonded to the upper surface, each
aligned in pairs with the section’s rib. The fully assembled
demonstrator with the hihglighted components is shown in
Figure 10. The morphing wing section is also fitted with two
DS3218 Digital RC 20 KG servo motor for actuation of the
bistable element in each rib. The bistable element features a
pair of actuation control horns. Nylon wires are attached
from holes in the control horns to the servo arms, forming an
antagonistic force pair. Rotation of the servo arm applies a
tensile force in one wire of the pair, providing the force
necessary to switch the bistable element through from one
stable state to the other.
Actuation tests (methods 2 and 3)
An experimental setup was designed to evaluate the
power consumption of actuation Methods 2 and 3 discussed in the previous section (Figure 11). This system
consists of a control interface to send command signals to
the servo motors, a programmable power supply, and the
ATM which acts as a probe to measure the trailing edge
deflection. The control interface utilizes the LINX
software package written for LabVIEW to operate an
Arduino UNO micro-controller in real time which sends
pulse-width modululation signals to command the positioning of the servos. The power for the servos is drawn
from a Keysight E36313 A power supply. The voltage of
the power supply is set at a constant 6 V, and the current
Figure 10. Manufactured wing section with two optimal
morphing rib topologies, composite laminate skin with
embedded double walled corrugation on the lower surface, four
MFC smart actuators, and two servo motors.
output is read directly by the LabVIEW interface using
the available software driver for the E36312 A power
supply. The product of these two values provides a
measure of the instantaneous power consumption of the
servos.
In order to compare the actuation Methods 2 and 3,
a baseline trailing edge deflection is established by
switching the bistable elements of the morphing ribs into
State 2 and relieving any tension on the nylon wires. This
baseline deflection was 22 mm and corresponds to the
state induced deflection of the wing section demonstrator.
The deflection is measured by lowering the cross-head of
the ATM until it barely makes contact with the trailing
edge and produces a force. For Methods 2 and 3 (Figures
11(b) and (c), respectively), the servo position is increased incrementally until a trailing edge deflection
equivalent to that of the State 2 equilibrium is measured
with the ATM. This position is recorded and used in an
automated control program to consistently deflect the
structure to the same location. Each actuation method is
subsequently tested five times. Every run of the test is
started at 15 s on the run timer. The control program then
ramps the output signal to the hold location. In the case of
the state switch motion, this involves positioning the
servo rotation to an angle that induces the arch evertion of
the bistable elements, holding that position for 5 s, before
moving to a position that relieves tension on the nylon
wires. Conversely, for Methods 2 and 3, the program
holds the servos for 60 s at the designated hold location,
then the servos return to the start position. The instantaneous angular position of the servo is illustrated by the
black dashed line in Figure 12(a), (b), (c), while the
power consumed at each instant is represented by multicolored solid lines (each color representing the Nth
repetition of each test).
The results indicate that Method 2 requires the least
initial energy input to reach the hold position (22 mm).
However, the nature of the servos requires that they
Rivas-Padilla et al.
747
Figure 11. Experimental servo actuation tests: (a) The experimental setup showing the components used, (b) The configuration of the
demonstrator and servos for the Method 2 actuation, and (c) The configuration of the demonstrator and servos for the Method
3 actuation.
periodically check their current position relative to the
command signal. Because the servos are holding against a
constant elastic strain in the structure, they are continually
being displaced from the target position. This causes the
continuous peak fluctuations observed in Figures 12(a) and
(b) after approximately 25 s. In contrast, the control horn
actuation (Method 3) shows a higher initial power draw,
relative to Method 2, both to deflect the structure and to
maintain the target deflection (Figure 12(b)). Note that the
Method 3 target deflection is achieved with a servo input
angle of 57°. Method 3 begins to outperform Method
2 when the input angle of the servo is increased to 90° and
then rotated back to 20° (Figure 12(c)). The power response shows the same initial peak as the one observed in
Figures 12(b) and (a) second peak (at around the 20 s time
mark) as the servo is rotated back to the at-rest position
(20°). This is likely due to a relatively quick servo
movement over a wide arch triggered by the control
program. The key advantage of this actuation strategy is
that the power consumption, after the bistable elements are
switched to State 2 and tension on the nylon wires is
released, is almost negligible. Some small peaks can be
seen, due to small positional error corrections of the servo,
caused from set point under- or overshooting.
By integrating the servo’s instantaneous power consumed over the test period, the cumulative work performed by the servo motors can be calculated to compare
the three methods tested (Figure 12(d)). In the case of the
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Journal of Composite Materials 57(4)
Figure 12. Test results showing the effects of different modes of actuation on power consumption. Top: curves show the instantaneous
power (left y-axis), in Watts, used to power the servo motors which hold a given angular position (right y-axis) for equivalent tip
deflections achieved by (a) Method 2: actuation directly at the trailing edge with a rigid control rod, (b) Method 3: actuating the control
horn without switching states, and (c) switching from State 1 to State 2. Bottom: (d) The curves indicate the cumulative energy consumed
by Methods 2 and 3, compared to the work required to switch the bistable element to State 2. The shaded regions indicate a range of
break-even points comparing the respective holding methods to switching states.
state switching (i.e., Method 3 with an increased input of
90°), only the initial energy necessary to switch the states
is determined, since the power consumption after that
point is insignificant. These state switching actuation
values provide the baseline (dashed colored lines) to
compare the other two methods. The Method 3 results are
represented by the dotted lines while Method 2 test results
are represented by the solid lines. A range of break-even
bands (blue for Method 3 and green for Method 2) can be
determined for both methods, when compared to the
baseline energy requirement of switching states. Specifically, the intersection of the plots corresponding to the
cumulative work increment of Method 2 (solid lines) and
3 (dotted lines) with the total work require to switch states
(dashed lines) are indicative of how long it takes before
it is more convenient to simply switch states to hold
the target deflection. For Method 3, this range is approximately 23–42 s while the break-even time interval
for Method 2 is 6–12 s. The lower end of the smaller
range (6 s) is even on the same length scale as the time it
takes to switch states. These key results show that integrating a bistable element into the design of a compliant
morphing structure provides an avenue to overcome the
elastic spring-back energy necessary to sustain target
deflections.
Wind tunnel tests of the wing section
We conduct wind tunnel tests to evaluate the aeroelastic
performance of the morphing wing section under aerodynamic loading. The wind tunnel tests are performed in
the Boeing Wind Tunnel at Purdue University (Figure 13).
The facility is equipped with a platform balance with three
load cells capable of measuring lift and rolling moments, as
well as a load cell for measuring drag. The wind tunnel is
also fitted with a LabVIEW data acquisition system to log
lift, drag, rolling moment, and wind tunnel velocity data.
A secondary LabVIEW VI is used for the actuation controls of the bistable element and to supply the necessary
voltage to the MFC actuators in order to morph the trailing
edge about each stable state. The details of the wind tunnel
component specifications, sensors, and systems are included in Ref. 54 The free stream velocity was set at 15 m/s
and 28 m/s and the aerodynamic performance data was
logged as the wing section model angle of attack is swept
from an angle of attack of 8° up to +8°. A grid paper with
Rivas-Padilla et al.
749
Figure 13. Boeing Wind Tunnel testing setup: LabView VI setup controls and wing section pitch while capturing Lift and Drag data from
the load balance table. A DSLR camera system captures the trailing edge displacement of the wing section surface.
4 mm squares is adhered to the back end cap near the
trailing edge of the demonstrator to track with a Canon
EOS Rebel T6i DSLR camera and an EF-S 18–135 mm
lens the displacement of the trailing edge at specific instances: State 1, State 2, and State 2 under aerodynamic
loads. Additionally, an actuation study is conducted at α =
4° to quantify the instantaneous power draw from the
servos and observe the lift transition between states as a
function of the servo angle.
The wind tunnel test is used to gather initial performance data of the morphing wing section at a wind speed
of 15 m/s when sweeping the wing section in both stable
states from 8° to +8° in steps of 2° (Figure 14(a)). The
results show a lift coefficient increase between 0.16 and
0.18 when switching from State 1 to State 2. This lift
increase is equivalent to pitching the airfoil in State 1
(i.e., undeformed configuration) by about +4°. The lift
increase is consistent throughout the range of tested
angles of attack, pointing towards the capability of the
morphing wing section to hold the deflected configuration
at and against different aerodynamic load cases. The CL
and CD polar results show a slight shift of 0.011 in the
positive drag direction (Figure 14(b)) when both ribs of
the wing section are switched to State 2. This horizontal
shift in drag is consistent with inducing camber to the
airfoil.
The aeroelastic response of switching between State
1 and State 2 at an α = 4° was captured using the DSLR
camera setup (Figure 15). Figure 15(a) shows the demonstrator in State 1 and Figure 15(b) shows the state induced
deflection of 22 mm achieved by switching from State 1 to
State 2. This significant morphing deflection is held without
the need of additional servo actuation power, i.e., no load is
sustained by the actuation at State 2. When the wind tunnel
speed is increased to 15 m/s, the aerodynamic loads push
Figure 14. Wind tunnel results: (a) Drag polar comparison
between State 1 and State 2 and (b) Lift variation between stable
states via state induced camber morphing.
back the trailing edge by about 2 mm, for a total of 20 mm of
downward deflection. This passive push-back deflection
was expected as it is accounted for during the weakly
coupled aeroelastic loop part of the optimization.
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Journal of Composite Materials 57(4)
Figure 16. State switching actuation test result showing (a) lift
increase as a function of servo angle increase (blue) and
decrease (red) and (b) average and peak power increase (blue
and green respectively) and decrease (magenta and red
respectively) as function of servo angle when switching from
State 1 to State 2.
Figure 15. (a) Side view showing the demonstrator mounted in
the wind tunnel at α = 0° and (b) trailing edge displacement
results at α = 4° showing stable state induced deflection at 0 m/s
and 15 m/s. The aerodynamic load pushes back the trailing edge by
about 2 mm.
A servo actuation study is performed to quantify the lift
variation and the servo’s instantaneous power draw at each
instant as the angle is increased and decreased with a wind
speed of 15 m/s. Figure 16(a) shows the servo angle increase
and corresponding CL increase induced by the tethered
nylon wire pulling on the control horn of the bistable element. The CL starts at 0.2 since the actuation study is
performed at α = 4°. The servo angle is swept from 0° to
100° (referred to as “Ramp Up”) increasing the CL as the
element switches from State 1 to State 2. The CL reaches a
maximum value of 0.36 when the servo actuation angle is at
100°. As the servo angle is decreased (referred to as “Ramp
Down”), the maximum CL slightly reduces to about 0.34,
and it remains at this value when the servo is reset back to its
initial position. The average and peak power draw of the
servo is also measured during the “Ramp Up” and “Ramp
Down” phases (Figure 16(b)). In the “Ramp Up” step, the
instantaneous average and peak power reach a maximum
value at an actuation angle of 40°, and then, the power draw
reaches a minimum at about 75° before starting to increase
again to a maximum value at an actuation angle of 100°.
This behavior highlights the power consumption as the
element switches from one state to rest at a second,
i.e., when the servo actuation angle reaches approximately
75°. The “Ramp Down” phase is characterized by a sharp
decrease in power consumption, reaching an average value
of 0 W when the servo actuation angle is at 70°. The
hysteretic behavior observed during the “Ramp Down”
phase clearly demonstrates how an actuator (in this case the
servos) do not need to exert additional effort once the tested
wing section “locks-in” onto the second stable state under
aerodynamic loads.
Rivas-Padilla et al.
A final higher speed test is performed at a wind speed of
28 m/s. This wind speed is lower than the target optimization speed of 35 m/s due to experimental limitations of the
wind tunnel structure. The demonstrator was switched from
State 1 to State 2 with an inclination of α = 6°. At this higher
speed, the model showed no indication of dynamic instability phenomena, highlighting the load carrying capability
of our demonstrator.
Conclusions
In this paper, we present an aero-structural optimization
approach for the design of morphing structures with selective stiffness and shape “lock-in” capability from embeddable bistable elements. To achieve this, we introduce
optimization objectives and a topology generation methodology. The optimization yields a structure capable of
optimally performing at two distinct flight conditions, while
satisfying the established lift constraints and maximizing
the lift ratio between stable states. The type of morphing ribs
obtained with this approach provide a potential avenue to
address the morphing structures trilemma, since the structure’s stiffness is adjusted to be more flexible when higher
deflection is needed, i.e. at lower flight speeds, reducing the
actuation requirements at this flight condition. In turn, the
reduced power would require less actuation weight, thus
limiting the penalties inherent to deforming a compliant
structure, compared to established mechanism-based control surfaces. The numerical aerodynamic performance of
the bistable rib topology is evaluated at each statically stable
state using a weakly coupled 2D aeroelastic loop, demonstrating the increase in lift-to-drag ratio via state induced
camber morphing, while at the same time reducing the
actuation requirements to control the trailing edge deflection
around State 2. This optimized rib topology is manufactured
and structurally tested showing good agreement between
experimental and numerical results. A numerical study was
presented comparing three modes of actuation and their
corresponding energy costs. It was observed that the MFC
actuators severely under-performed the other two actuation
methods since they required 20 times more energy to morph
the rib to the optimal target deflection. This indicates that the
MFCs surface mounted actuation might not be able to
exhibit sufficient mechanical advantage or coupling to
compliant modes to effectively deform structurally efficient
semi-monocoque (rig-supported) morphing structures.
The optimized rib topology was used to manufacture a
morphing wing section with two bistable ribs. An experimental set of servo actuation tests were performed on the
demonstrator to evaluate the power requirements of actuating the compliant morphing system. When considering the
cumulative work required, the energy cost to switch states
outperforms actuation Method 3 when the deflection needs
to be sustained for over 45 s or more given the continuous
751
power draw to resist the elasticity of the compliant demonstrator. Although, Method 3 requires less energy than
Method 2, the added structural stiffness of the rigid rod from
Method 3 restricts the state induced camber morphing effect
when switching to State 2. Preserving this shape “lock-in”
feature of bistable structures is crucial to take full advantage
of its nonlinear properties. These results motivate future
investigation to determine actuation paradigms enabling the
designer to take full advantage of the interaction between
bistable structures and actuators.
The morphing wing section was tested experimentally in
a sub-sonic wind tunnel at flight speeds of 15 m/s and 28 m/
s. This initial testing campaign demonstrated the lift variation capability achieved by switching between stable
configurations, while no dynamic instability was triggered
when actuating the system at the higher aerodynamic loads.
Finally, the trailing edge displacement measurements show
that the trailing edge deflection can hold its deflected
configuration even when exposed to aerodynamic loads.
This work has provided additional evidence that exploiting
instabilities in compliant structures can yield potential
avenues for improving the performance of shape adaptable
systems.
Acknowledgements
The authors would like to acknowledge the assistance of Prof.
Sally Bane in the management and use of the Boeing Wind Tunnel.
Finally, credit for work on the design of the corrugation must be
given to Liang Yang during his tenure at the Programmable
Structures Laboratory.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
This work partially supported by the the US Air Force Office of
Scientific Research (AFOSR) under the Grant FA9550-17-1-0074
“On-demand Stiffness Selectivity for Morphing Systems.” J.R.
Rivas-Padilla and Andres F. Arrieta were supported by this grant.
D. Matthew Boston was partially sponsored by the Army Research
Laboratory and was accomplished under Cooperative Agreement
Number W911NF-16-2-0008.
Disclaimer
The views and conclusions contained in this document are those of
the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research
Laboratory or the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein.
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Journal of Composite Materials 57(4)
ORCID iD
José R Rivas-Padilla  https://orcid.org/0000-0002-8010-2482
14.
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Appendices
Appendix A: Double-walled
corrugation homogenization
The in-plane extension and compression of the lower rib
skin surface is crucial in the design of the camber morphing
rib. In this work we use a double-walled corrugation described in detail by Ref. 31, to achieve the necessary
compliance of the lower surface of the rib. The double-wall
corrugation structure is constructed from four unit corrugation topologies. The dimensions of each unit were chosen
such that a set of four-unit corrugations could be 3D printed
together and fitted in the limited space within the morphing
rib beams (Figure 17(a)). The lateral wall thickness of each
unit (0.6 mm) was limited by the 3D printer capabilities of
the Ultimaker 5S printer and the thermoplastic polyurethane
(TPU) printing parameters. The top and bottom wall
thickness (2 mm) were selected to decouple as much as
possible the in-plane stretching and out-of-plane bending,
since the decoupling depends on a high ratio between top/
bottom and lateral wall thickness as per the findings presented in Ref. 31 Similar dimensions of the unit corrugations were also used in Ref. 52 to achieve the necessary
camber morphing behavior.
The homogenization of this double-walled corrugation is
crucial to model a smooth continuous lower surface of the
rib. Without this continuity of the surface, it would not be
possible to run the 2D aeroelastic analysis tool using the
XFLR5 (XFOIL) aerodynamic analysis software. In practice, it is also necessary to cover up this discontinuity in a
compliant morphing wing to avoid drag penalties. However,
addressing this challenged is left as a necessary step in the
future work of this project.
A beam element structure of the entire corrugation is
modeled in ABAQUS with the left end of the corrugation
pinned in the x-y plane. The boundary condition of the right
end depends on whether the corrugation is to be subjected to
a pure stretching deformation (Figure 17(b)) or pure
bending deformation (Figure 17(c)). The relative thickness
of each section of the beam model has been rendered for
visualization purposes. The stiffness matrix relationship for
a shell element undergoing deformation in only two dimensions can be simplified to:
A B ϵ
N
¼
(11)
B D κ
M
Journal of Composite Materials 57(4)
where A is the stretching stiffness, B is the stretchingbending coupling stiffness, D is the bending stiffness, ϵ is
the strain and κ is the curvature of the corrugation. For the
pure stretching case, the right end of the corrugation is
modeled as a slider and both ends are restricted to a zeroslope boundary condition. Given these boundary conditions, κ = 0 at both ends, and the equations for the force and
moments, simplify to:
A*ϵ ¼ N
(12)
B*ϵ ¼ M
(13)
The strain, reaction forces (N) and moments (M) can be
extracted from the numerical model to calculate the in-plane
stretching stiffness component A, and the stretchingbending component B. For the pure bending case, the left
end is pinned, and a concentrated moment is applied at both
ends. We assume the transverse (y-direction) displacement
of the corrugation to be small enough, such that the induced
strain is ϵ ≈ 0. We can check this assumption by extracting
the reaction forces and rotations at the end points of the
corrugation to calculate again the stretching-bending coupling B using:
B*κ ¼ N
(14)
where κ is the corrugation curvature. The curvature is
calculated using:
κ ¼ ðURz, P1 URz, P2 Þ=l
(15)
where URz,P1 and URz,P2 are the rotations about the nodes at
pin 1 and pin 2 respectively, and l is the length of the
corrugation.
It is observed that both stretching-bending stiffness
components calculated from each analysis are within 0.32%
difference, validating the small strain assumption. Finally,
the bending stiffness coefficient D is calculated by extracting the rotation and reaction moment values at the end
points of the corrugation and using:
D*κ ¼ M
(16)
The obtained stiffness coefficient values were A =
2.17 N, B = 12.51 N-mm, and D = 259.16 N-mm2. These
values correspond to shell corrugation width of 40 mm.
Thus, each stiffness component must be divided by the
length (40 mm) to obtain the stiffness values per unit width
of the corrugation. The general shell stiffness feature is used
in ABAQUS to model the corrugation section of the
morphing rib and the stiffness properties are specified using
the calculated A, B, and D stiffness values per unit width
from the homogenization model proposed. The rest of the
stiffness properties are assumed to be orders of magnitude
Rivas-Padilla et al.
755
Figure 17. Double-walled corrugation: (a) Unit and expanded structure dimensions, (b) boundary conditions for pure stretching
analysis, and (c) boundary conditions for pure bending analysis.
Figure 18. General shell stiffness values specified for the homogenization of the double-walled corrugation.
stiffer (104) and all the coupling coefficients, except for the
stretching-bending stiffness are set to 0 (Figure 18).
Appendix B: Numerical model details of the single
rib model
Figure 19. Morphing rib section with color coded regions
identifying each component in the model.
The morphing rib modeled is divided into multiple sections
(Figure 19) with a specific combination of thermoplastic
756
Journal of Composite Materials 57(4)
Table 2. Material layup and thickness for each section of the numerical model.
Section Description
Material layup
Spar [blue]
Upper-Skin + MFC [Purple]
Lower-skin [cyan]
Rib [green]
GBS element [red]
Double-wall corrugation [yellow]
Onyx [4.2 mm]
Onyx [1.2 mm]+[0, 0/90 pw, 0] laminate [0.386 mm] + MFC [0.3 mm]
Onyx [1.2 mm]
Onyx [1.2 mm]
Onyx [0.8 mm]
General shell stiffness (Figure 18)
polymer materials, composite laminate, and smart actuators detailed in Table 2. The properties of the MFC actuators are from a single crystal PMN and the piezoelectric
behavior is modeled using the thermal analogy described
in.55 These types of actuators were selected due to their
multi- functional nature as they serve both as actuators and
structural elements, while adding little weight to the
morphing wing structure. The mesh for the model is
constructed with four-node, doubly curved, thin shell,
reduced integration, linear shell elements (S4R). The
ABAQUS/Standard module is used to conduct the analysis. All loading boundary conditions are assumed to be
applied in a quasi-static manner. A general static analysis is
therefore considered sufficient for each step. The approximate
element size for the spar, compliant rib, and corrugation is
2.5 mm while the approximate element size for the GBS
element is 1.00 mm to accurately capture the more complex
stress field at the regions of high strain. The total number of
elements for the assembly is 13,416. This relatively fine mesh
is appropriate due to the large nonlinear geometric displacements occurring particularly around the curved arch of
the bi-table elements. The nonlinear geometry solver,
“Nlgeom,” option is also turned on for this reason. The
structure exhibits a negative stiffness and release of strain
energy when transitioning between stable solutions. The
ABAQUS documentation recommends the addition of artificial numerical damping to solve geometrically nonlinear
static problems involving buckling or snap through of a
structure56 A small amount of numerical damping on the
order of 107 is therefore used throughout the analysis.
The rib model is fixed along the spar in the initial analysis
step. The bi-stable element is then pinned at the four corners
of the curved arch region and a prescribed-displacement
boundary condition (20 mm) is applied to evert the arch to
the second stable state. The pinned support and prescribeddisplacement boundary conditions are deactivated in the
third step of the analysis. This allows the structure to relax
and freely transition into its stable configuration. The final
step then introduces a “perturbation” displacement
boundary condition at the trailing edge of the wing section.
This is done to measure the reaction force at the point of
displacement and characterize the global stiffness behavior
Figure 20. Initial set of 5 topologies to initiate the GA
optimization loop. The bistable element location is highlighted in
red.
Figure 21. Schematic showing location and labels of structural
nodes for the optimized rib geometry. Red nodes correspond to
the lower surface nodes (LSN), green nodes correspond to the
mid line nodes (MLN), and yellow nodes correspond to the upper
surface nodes (USN) of the rib geometry.
Table 3. Chordwise node location for optimal rib geometry.
General node location
Label
Dimension
Units
Lower surface node 1
Lower surface node 2
Lower surface node 3
Mid line node 1
Mid line node 2
Upper surface node 1
Upper surface node 2
Upper surface node 3
LSN1
LSN2
LSN3
MLN1
MLN2
USN1
USN2
USN3
146.74
181.51
366.46
178.60
361.39
181.51
329.75
364.33
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
Rivas-Padilla et al.
about each stable state when the trailing edge is deflected
downward. In the case of the 2D aeroelastic loop, this final
step is replaced with an aerodynamic loading step. The node
coordinates of the airofil profile are extracted and the shape
of the airfoil is provided to the XFLR5 aerodynamic
analysis tool to calculate the presssure coefficients (Cp)
along the upper and lower surface of the rib. This pressure
distribution is then imported back into ABAQUS and integrated over the area of the rib to calculate the aerodynamic
load distribution to model the aeroelastic response of the
compliant rib.
Appendix C: Initial genetic algorithm
population individuals
The genetic algorithm evaluates 100 individuals at each
generation. To improve the convergence speed of the optimizer, 5 suboptimal topologies were obtained from prior
optimization runs. The inclusion of these individuals in the
optimizer yield rib geometries with satisfactory morphing
results and aerodynamic properties (Figure 20). In particular, the first three optimization runs were conducted by
seeking a target lift coefficient with maximum stiffness
change by allowing the optimizer to assign “zero stiffness”
to one of the truss members, i.e., removing one element.
This approach was also used to determine the solver’s
sensibility to the most extreme case of stiffness change. The
optimizer was then modified to include the geometry and the
nonlinear structural response of the GBS element. This
modification enables us to study both the selective stiffness
behavior and the shape locking effect achieved by storing
757
strain energy via the state switching of the GBS element.
Once the GBS element response is included into the optimizer, instead of removing a truss element to simulate the
stiffness change, the element is replaced by the GBS element. Topologies four and five are suboptimal solutions
generated from this evolution in the optimization method.
The other 95 individuals in the initial population are randomly generated by the the optimizer to initiate the optimization and obtain an optimized geometry. The relative
location of the structural nodes corresponding to the optimized rib geometry presented in this paper are shown in
Figure 21 and Table 3 shows the exact chord-wise location
distances of each node with respect to the leading edge of
the rib geometry.
Cd,flex
Cl,flex
Cd,stiff
Cl,stiff
CL
CD
fi
gi
Pf
Vstiff
Vflex
V∞
αi
f
α
δx
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2D drag coefficient of State 2 (flexible)
2D lift coefficient of State 2 (flexible)
2D lift coefficient of State 1 (stiff)
2D lift coefficient of State 1 (stiff)
3D lift coefficient for the wing section
3D drag coefficient for the wing section
Objective functions
Constraint functions
Penalty function
State 1 Actuation Voltage
State 2 Actuation Voltage
Free stream velocity of wind tunnel [m/s]
Scaling factors of objective function
Pseudo objective function
Angle of attack [°]
Change in length of GBS element [mm]