italian association of aeronautics and astronautics xxii
TRANSCRIPT
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Italian Association of Aeronautics and Astronautics
XXII Conference
Napoli, 9-12 September 2013
STRUCTURAL ANALYSIS OF THE FISH BONE ACTIVE CAMBER
CONCEPT
B.K.S. Woods1*
, M.I. Friswell1
1College of Engineering, Swansea University, Swansea, UK
ABSTRACT
Currently, a biologically inspired airfoil camber morphing mechanism known as the Fish
Bone Active Camber is under development. Initial prototypes have shown a large control
authority and significantly improved aerodynamic performance over traditional trailing edge
flaps. Further advancement of the concept requires a structural analysis method which can
accurately capture the stiffness properties and large deformations of the compliant morphing
structure. Additionally, it is desired that this code be easily updated for changes in model
geometry and material properties and that the computational costs be minimized. This work
presents two methods; one a low-fidelity approach based on Euler-Bernoulli beam theory,
and the other a high fidelity commercial Finite Element Analysis software. Deflection
predictions from both methods are compared to experimental results for a Fish Bone Active
Camber prototype. The Euler-Bernoulli beam theory based analysis is found to provide good
levels of accuracy at very low computational cost. While the Finite Element Analysis
software is significantly more accurate for some, but not all, of the configurations tested, it
always takes significantly longer to run. The beam theory analysis is also more amenable to
changes in geometric and material properties, and so is the focus of continued work.
Keywords: morphing, camber, airfoil, structures
1 INTRODUCTION
The use of geometric and material compliance to create morphing airfoil mechanisms requires
careful consideration of the methods used to model the structural behaviour. Given the
complex nature of the geometries typically used to provide sufficient compliance to create
significant changes in airfoil geometry, many previous researchers have used high fidelity
Finite Element Analysis (FEA) as the primary means of structural analysis.[1-8] While FEA
methods are generally capable of analysing any given geometry (within certain limitations),
they require significant care and effort to integrate into higher levels of analysis code, for
instance a geometry optimization code, particularly when the design is desired to be varied
over a wide range of parameters. Furthermore, FEA codes generally require a significant
amount of computation time to solve, a situation made considerably worse when the
deflections considered are large. Lower fidelity structural analysis methods are available
which run much quicker and are easier to integrate with other codes, but often they do not
provide the desired level of accuracy. This disparity between accuracy and computational
efficiency is a common and widely experienced problem. This work considers the impact of
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this question on the analysis of a specific active camber airfoil morphing mechanism which is
currently under development.
1.1 Fish Bone Active Camber concept
The Fish Bone Active Camber (FishBAC) mechanism provides a novel means of generating
large, bidirectional changes in airfoil camber. Introduced by Woods and Friswell[9], this
concept employs a biologically inspired compliant structure to create large, continuous
changes in airfoil camber and section aerodynamic properties. The structure consists of a thin
chordwise bending beam spine with stringers branching off to connect to a pre-tensioned
Elastomeric Matrix Composite (EMC) skin surface. Both core and skin are designed to
exhibit near-zero Poisson‟s ratio in the spanwise direction. Pre-tensioning the skin
significantly increases the out-of-plane stiffness and eliminates buckling when morphing.
Smooth, continuous bending deflections are driven by a high stiffness, antagonistic tendon
system. Actuators mounted in the D-spar drive a tendon spooling pulley through a non-
backdrivable mechanism (such as a low lead angle worm and worm gear). Rotation of the
pulley creates equal but opposite deflections of the tendons. These differential displacements
generate a bending moment on the rigid trailing edge strip, which then induces bending of the
trailing edge morphing structure to create large changes in airfoil camber. A schematic
overview of the FishBAC concept is shown in Figure 1. Since the tendon system is non-
backdrivable, no actuation energy is required to hold the deflected position of the structure,
reducing control action and power requirements. Furthermore, the automatic locking action
of the non-backdrivable mechanism allows the stiffness of the tendons to contribute
significantly to the chordwise bending stiffness of the trailing edge under aerodynamic load,
without increasing the amount of energy required to deflect the structure.
Figure 1 Fish Bone Active Camber concept
Wind tunnel testing of the prototype seen in Figure 2 found the FishBAC provided
improved aerodynamic efficiency compared to traditional trailing edge flaps, with increases in
lift-to-drag ratio of 25% being realized at equivalent lift conditions.[10] An increase in lift
coefficient of ∆Cl = 0.72 between unmorphed and morphed was measured at a freestream
velocity of V∞ = 20 m/s and an angle of attack of α = 0°.
The large achievable deflections and continuous compliant architecture make this
concept universally applicable to fixed wing applications ranging in scale from small UAVs
to commercial airliners, and to rotary wing applications including wind turbines, helicopters,
tilt-rotors, and tidal stream turbines.
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Figure 2 FishBAC wind tunnel test model showing deflected shape [10]
1.2 Objectives of current work
The objective of this work is to compare the effectiveness of low and high fidelity structural
analysis methods for capturing the stiffness properties of the FishBAC. Ongoing work on this
concept requires a structural solver for use in a highly coupled Fluid-Structure-Interaction
(FSI) code running inside of optimization routines. To this end, the structural partition of the
FSI code must be able to easily accommodate changes in geometry and material properties. It
also must be able to solve for the deflected state of the FishBAC under internal (e.g. actuator)
and external (e.g. aerodynamic) loading with minimal computational cost. Solution times on
the order of a few seconds or less are desired. Finally, the structural analysis method must
provide sufficient accuracy to be useful as a design tool. To put some rough quantitative
guidelines on the accuracy requirements, for this work less than 10% error is considered
necessary and less than 5% would be highly desirable. The accuracy of the two methods will
be assessed by comparing them to the results of four different experimental tests. Two
different geometric configurations, FishBAC core only and FishBAC core with the skin
bonded on, will be tested under two loading conditions: externally applied tip loads and
bending moments created by the drive tendons.
2 STRUCTURAL ANALYSIS METHODS
The FishBAC structure was analyised using two methods with different levels of fidelity. A
low fidelity model was devised based on the Euler-Bernoulli beam theory. This method is
simple, robust, very quick computationally, and intrinsically parameter driven. A high
fidelity Finite Element Analysis software, namely SolidWorks Simulation 2013, was also
used to give a counterpoint on the computational cost versus accuracy scale.
2.1 Low fidelity Euler Bernoulli beam theory analysis
This section introduces a low fidelity structural model developed for the FishBAC based on
Euler Bernoulli (EB) beam theory. This model allows for very quick analysis of design
configurations and for parameter driven modifications to the geometry and material
properties. The derivation of the analytical formulation for FishBAC stiffness from Euler-
Bernoulli beam theory will be shown, the boundary conditions will be presented, and the
distribution of flexural rigidity along the chord will be formulated. The bending moment
produced by the tendons is included into the integration of the Euler-Bernoulli equations.
2.1.1 Governing Equations
The low fidelity structural model is analytical and derived from Euler-Bernoulli beam theory.
The low bending stiffness and high length to thickness ratio of the bending spine, low in-
plane stiffness of the skin, and the continuous loading along the span potentially make Euler-
Bernoulli theory a good initial approximation for analysis. Furthermore, the high stiffness
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stringers branching off from the spine help to enforce the „plane sections remain plane and
normal to the neutral axis‟ assumption which underlies the derivation of Euler-Bernoulli beam
theory.[11]
The bending deflections of the morphing structure are found by integrating the
aerodynamic pressure distribution to find shear distribution, integrating that to find moment
distribution, integrating again to find the distribution of slope, and then integrating one final
time to find the deflection of the neutral axis
The relationship between net aerodynamic pressure, p, flexural rigidity, EI, and
vertical displacement, w, is given by the Euler-Bernoulli beam equation:[11]
(
) (1)
Integrating pressure produces force, specifically the vertical shear force acting on the beam:
(
) ∫
(2)
The integral of shear force is bending moment:
∫
(3)
The curvature of the spine can be found by rearranging Equation 3:
(4)
Curvature is then integrated to give slope:
∫
(5)
And finally, integrating slope provides the distribution of vertical deflection:
∫ (6)
While this formulation can be seen to be derived for a FishBAC operating under
aerodynamic pressure loading, it is easily adapted to the tip load and tendon moment loading
cases studied here by setting the applied pressure distribution to p(x) = 0 and by changing the
boundary conditions, as will now be discussed.
2.1.2 Boundary Conditions
The integration constants which are generated during the solution of the Euler-Bernoulli
equations are solved for by considering the boundary conditions. For the FishBAC, the
bending spine is assumed to be clamped at its attachment to the rigid D-spar. The other
boundary conditions are then dependent on the loading condition being studied. In this work,
two different loading conditions are analyised, a tip load and a tendon moment. The tip load
defines a shear force boundary condition at a point close to the trailing edge. In the test setup
used, this load cannot physically be applied to the extreme end of the trailing edge, and so a
loading position, xload, slightly inboard from the trailing edge is defined.
The tip load boundary conditions are therefore:
(7)
(8)
Similarly, for the tendon moment loading condition, a prescribed moment is applied at
the anchor point of the tendons, xten, leading to the following boundary conditions:
(9)
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(10)
2.1.3 Flexural Rigidity Formulation
The flexural rigidity (EI) distribution of the FishBAC structure is formulated as a linear
superposition of its components in this analysis. The spine is modeled as a constant thickness
beam, and since its neutral axis is coincident with that of the FishBAC structure as a whole,
the flexural rigidity of the bending spine, EIbs, is equal to:
(11)
where Ebs, is the elastic modulus of the bending spine material, b is the span of the FishBAC
segment and, tbs, is the thickness of the bending spine. Note that the rigidity of the spine
could be made to vary along the chord without changing the formulation or implementation of
this analysis. Indeed, tapering the thickness or modifying the material properties of the spine
is an effective means of controlling the deflected shape, and therefore aerodynamic properties,
of the FishBAC. While the results presented here are for a constant thickness spine, current
work by the authors seeks to exploit these design variables to optimize the structure for
various performance metrics.
The skin has a low inherent flexural rigidity, but because it is attached at a
considerable distance from the neutral axis of the FishBAC, its contribution to overall
stiffness must be considered. The flexural rigidity of the skin on the FishBAC, EIsk, is
therefore modeled using the parallel axis theorem.[12] The upper and lower skins are
considered separately and then added:
*
+ *
+ (12)
here Esk, is the elastic modulus of the skin material and, tbs, is the thickness of the skin, rl, is
the distance between the lower skin surface and the neutral axis and, ru, is the equivalent
distance for the upper skin surface. As with the spine, the skin in this initial analysis is a
constant thickness along the chord. However, in this case the flexural rigidity in not constant
with chord due to the changing thickness of the airfoil. Significant reductions in stiffness, and
therefore actuation requirements, could likely be obtained with no detrimental effect on
maximum out-of-plane displacement if the skin thickness were tapered to match the
distribution of aerodynamic pressure coefficient. Due to their thinness in the chordwise
direction, the stringers are assumed to have a minimal impact on global bending stiffness, and
are therefore are not included in the structural rigidity of the FishBAC for this initial analysis.
The total FishBAC flexural rigidity, EItot, is the linear sum of the spine and skin rigidities:
(13)
In order to model the experiments presented below in which only the FishBAC core
was present (performed before the skin was bonded on), the elastic modulus of the skin, Esk¸
was set to zero.
2.1.4 EB analysis run time
The run time for this analysis when coded into Matlab R2012b engineering software is quite
low, requiring roughly 2 seconds on an Intel i7-3549M CPU with 16GB of RAM to perform
the required geometry and loading definition and numerically integrate the Euler-Bernoulli
equations. If the geometry is pre-defined and only the loads are changed for a given run, then
the solution time is even less, taking well under a second.
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2.2 High fidelity finite element analysis
A commercially available high-fidelity Finite Element Analysis (FEA) code was also used to
predict the stiffness properties of the morphing structure. Initially, both ABAQUS CAE 6.11
and SolidWorks Simulation 2013 were considered, with both programs producing nearly
identical displacement predictions during preliminary comparison. SolidWorks Simulation
was chosen for use during the remainder of this analysis as it directly integrates in with the
SolidWorks Computer Aided Design (CAD) software in which the FishBAC components
were designed. This allows for the exact geometry (including fillets, mounting brackets,
tendon holes etc) to be modelled, and it also greatly simplifies updating the analysis when
geometry or material properties are changed. Furthermore, SolidWorks has built in parameter
driven design tools and can be run from external macros, which means integration into
existing FSI or optimization codes is feasible.
2.2.1 Solution parameters
Both the core and the skin were modelled with solid tetrahedron elements, with a curvature
based mesh to more accurately geometric features such as fillets. Given the two-dimensional
nature of the loading in all configurations tested, the span of the model was reduced to
decrease solution time, and the loads scaled accordingly. The FEA model was 10 mm in
span, whereas the prototype built was 150 mm.
The analysis was performed using an iterative solver (FFEPlus) and large
displacement mode. This insured proper load tracking and geometry updating under the large
deformations experienced by the model, particularly in the core only configuration when the
stiffness is significantly lower. Convergence studies found that approximately 60,000
elements were needed for the FishBAC core to produce a converged solution, and 100,000
elements were needed for the core with skin bonded on.
2.2.2 Typical results
Representative results are shown below for the FishBAC skin and core under tip load. Figure
3a shows the distribution of von Mises stress. Note how the spine carries most of the bending
stresses. This is primarily due to the very large difference in stiffness between the skin and
core, as noted in Table 2. By truncating the scale of stresses shown, as in Figure 3b, we can
get a better idea of the variation in stress present in the skin and stringers. Here can be seen
interesting details such as the variation in skin stress along the chord and the local effects of
the bonding between skin and stringers. In this work, the required displacement results under
a given loading condition were found by probing the mesh at points matching the
experimental loading conditions.
(a)
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(b)
Figure 3. Typical FEA results showing, (a) full range of von Mises stress, and (b) truncated
range of stress to highlight varitation in skin and stringers
2.2.3 FEA run time
Due to the iterative solver used and large displacements, run time varied significantly with the
amount of force or torque applied. Lower load levels would take 5-10 minutes to solve on an
Intel i7-3549M CPU with 16GB of RAM, while the largest load levels tested would take a
little over 2 hours to solve. This is considerably longer than the roughly 2 seconds required
for a solution from the Euler-Bernoulli beam theory analysis.
3 EXPERIMENTAL METHODS
This section will present the various experimental tests conducted to validate the structural
models. This begins with the determination of the material properties of the acrylonitrile
butadiene styrene (ABS) plastic and silicone matrix, carbon fiber reinforce elastomeric matrix
composite from which the test article was made. Then the bending stiffness of the article
under two types of loading, tip load and tendon moment, was measured, both with and
without the elastomeric skin attached for a total of four experimental tests.
3.1 Material characterization
3.1.1 ABS Plastic Modulus
In order to determine the effective modulus of the ABS material used by the 3D printer, a thin
beam with constant cross section was printed and cantilever tested. The thickness of the beam
was set equal to the spine thickness of the FishBAC prototype to help minimize any changes
in properties due to scale. It is important to note that due to the finite size and circular shape
of the bead of plastic extruded by the printer nozzle, the block of material created is not fully
dense. Furthermore, there is directionality to the filling of the interior volume of the beam
which may influence the stiffness properties of the resulting structure. The geometric
parameters of the specimen tested are given in Table 1.
Table 1. ABS Plastic Modulus Test Specimen Parameters
Parameter Value
Length from clamped end (L) 159.01 mm
Thickness (t) 1.99 mm
Width (w) 25.31 mm
Loading point from clamped end (a) 148.50 mm
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The manufactured beam was 15 mm longer than the nominal test length to provide an
area for clamping. A clamped boundary condition was created by clamping the beam
between thick steel plates and to a table with a G-clamp, as seen in Figure 4. A ruler with 0.5
mm graduations was aligned vertically at the end of the beam such that the deformations
could be easily read. Given that the tip deformations of the beam were on the order of 30
mm, this measurement device provided sufficient accuracy for the purposes of this
experiment. A thin piece of Spectra cordage was taped to the top of the beam precisely at the
desired loading point, a, and then formed into a loop underneath the beam where masses
could be hung. The steel masses were individually weighed on a precision scale to
compensate for variances between their nominal and actual masses.
Figure 4: Experimental test setup for ABS plastic cantilevered beam testing
Euler-Bernoulli beam theory was also used to derive the elastic modulus of the ABS
plastic, EABS, from the experimental results. As this beam has a constant cross-section and
simple boundary conditions, an exact solution for the variation of tip deflection, wtip, with
applied load can be derived. This is given in [12] as:
()
()
The value of elastic modulus which provided the best fit to the experimental data was
then found using least squares regression. As can be seen in Figure 5, the matching achieved
was quite good, with a coefficient of determination, R2 = 0.9983, and an experimentally
determined modulus of EABS = 2.14 GPa. This value was used in both the low fidelity and the
high fidelity structural models.
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Figure 5. Comparison of experiment with Euler Bernoulli beam theory result
3.1.2 Elastomeric matrix composite elastic modulus
Reliable experimental measurements of the material properties of the silicone matrix, carbon
fiber reinforced EMC skin were not available for this study, and so the primary property of
interest, the elastic modulus in the morphing direction, was estimated using the high fidelity
FEA code and the experimental results. As will be shown, the FEA predicted very accurately
the stiffness of the FishBAC core by itself. By comparing these results to the skin and core
results, a value of skin modulus, Esk, which provided a similar level of accuracy for the FEA
predictions could then found. This calculated value was Esk = 4.56 MPa, which is of
reasonable magnitude compared to previous experimental work.[13] This value was also used
without modification in the EB analysis code.
3.2 FishBAC test article construction
In order to allow for experimental characterization of the FishBAC concept, a prototype test
article was designed and built. This test article does not incorporate all four aspects of the
concept in their entirety, as some simplifications were made in the interest of the goals of the
current effort; only the aspects directly related to the inherent structural stiffness properties
were included. Specifically, the model does not include a non-backdriveable mechanism, and
a single agonist tendon is employed instead of an agonist/antagonist pair. These
modifications make it easier to generate a known bending moment on the trailing edge by
applying a tension to the tendon which is mounted at a known moment arm from the neutral
axis. Furthermore, while the non-backdriveable mechanism element of the concept is seen as
a means of increasing the stiffness of the structure to external loading (in addition to
providing a zero power input position hold capability), the baseline stiffness of the structure
must first be understood without it. This particular element of the concept is still under active
development, and so future work will examine its contribution to stiffness and determine
appropriate modelling methods to capture its behaviour. The non-morphing leading edge of
the prototype has also been removed for this testing. It has been assumed for this analysis that
the leading edge D-spar is also not included in the present analysis. This component is
essentially rigid in the chordwise direction, and it is attached to the morphing section in a
manner which produces a clamped boundary condition. For the sake of the experiments and
analysis presented here, it was computationally, analytically, and mechanically simpler to
clamp the end of the morphing section directly instead of through the leading edge. In this
way, all three methods have the same boundary conditions, facilitating direct comparison. An
overview of the geometric parameters of the FishBAC prototype is presented in Table
2Error! Reference source not found..
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Table 2. FishBAC prototype parameters
Parameter Value
baseline airfoil NACA 0012
chord (c) 305 mm
span (b) 150 mm
start of morph (xs) 0.35c = 107 mm
end of morph (xe) 0.85c = 260 mm
spine thickness (tb) 2 mm
# of stringer pairs 14
stringer thickness (tst) 0.8 mm
skin thickness (ts) 1.5 mm
tendon offset (yt) 4.2 mm
tendon diameter (dt) 0.7 mm
core modulus (EABS) 2.14 GPa
skin modulus (Esk) 4.56 MPa
The structural components of the model, with the exception of the skin, were printed
from ABS plastic using an HP Designjet 3D Fused Deposition Modeling printer. Due to the
discrete nature of the printing process, this manufacturing method produced small difference
between the nominal dimensions presented in Table 2 and the manufactured article. While
small (on the order of 0.05 mm) these differences can have a significant impact on stiffness,
particularly in the case of spine thickness which has a highly non-linear impact on chordwise
bending stiffness. For this reason the analytical and FEA simulation results presented here
use the actual measured values.
The silicone matrix, unidirectional carbon fiber reinforced EMC skin was made using
a multi-step laminating process which was a modified form of that presented elsewhere. [13-
14]
3.3 Tip load testing
This section will describe the setup and testing of the FishBAC structure under tip load. This
test gives an idea of the basic bending stiffness under external loads. While it is not strictly
representative of the distributed aerodynamic pressure loading that will in reality act on the
FishBAC, it is a simple test with known boundary conditions and only a few parameters. This
makes it a good starting point for establishing the validity of the structural models.
The FishBAC prototype was clamped to a rigid table using large G-clamps and
aluminium flanges bolted to the plastic core of the FishBAC. The model was raised up off of
the surface of the table to provide space for downwards deflections, as seen in Figure 6. This
test applied known displacements to the tip of the trailing edge using a digital height gauge.
The resulting reaction load was measured with a strain gauge load cell mounted between the
tip of the height gauge and the trailing edge. A stiff aluminium spreader bar was attached to
the load cell to provide a distributed loading condition. Furthermore, a small diameter (3
mm) steel rod was attached to the face of the spreader bar such that there was only a single
line of contact between the displacement application apparatus and the trailing edge. As it is
not possible to apply displacement to the exact end of the trailing edge, a point slight inboard
(9.5 mm from the trailing edge) was marked and carefully aligned with the steel rod. This
position was also used in the models for the load application and deflection measurements.
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Figure 6. Tip load experimental setup (shown with FishBAC skin and core)
3.4 Tendon moment testing
The section will discuss the experimental setup of the second set of stiffness tests performed
on the FishBAC structure where the structure was deformed by internal tendon moments in a
manner closely approximating the actuation method for this concept.
For this experiment, the FishBAC prototype was clamped vertically in a vise. Tendon
moments were generated on the trailing edge of the FishBAC by attaching a Spectra fiber
tendon to the anchor point on the trailing edge and then threading it through the orifices in the
stringers until it left the FishBAC core at its leading edge extent. Masses were then
suspended from the exposed portion of the tendon to generate a known tension in the tendon.
This tension is converted into a moment using the known tendon mounting offset (yt = 4.2
mm). For this experiment, the effects of any interactions between the tendon and the stringer
orifices were ignored. Figure 7 shows the arrangement for this test; note that the tendon is not
quite visible behind the blue steel support structure.
Figure 7. Tendon moment experimental setup (shown with FishBAC core only)
4 RESULTS AND DISCUSSION
The results of the experiments and simulations for the four different configurations will now
be shown and discussed.
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4.1 Stiffness under tip load
We start with the simplest configuration considered, the tip load, core only test, the results of
which are shown in Figure 8. It can be seen that the experimental response is linear, with a
coefficient of determination, R2 = 0.9994, despite tip deflections which are quite large. The
maximum measured tip deflection of 40 mm is over 13% of the airfoil chord, and over 20% of
the length of the morphing section. This linear behaviour is also seen in the EB analysis code
results, which is expected given the linear nature of this formulation. The FEA also predicts a
linear response over this displacement range. In terms of prediction accuracy, it can be seen
that, as anticipated, the FEA shows excellent agreement with the experimental results under
this simple loading condition. The analytical code also shows quite good accuracy however,
and while under-predicting the stiffness slightly, it still displays a very useful level of
accuracy. Fitting linear regressions to both the experimental and EB analysis results gives
slopes (equal to the effective linear stiffness) that vary by only 5.8%.
Figure 8. FishBAC stiffness under tip load, core only
We next consider the same loading condition but with the skin bonded onto the core.
The results for this configuration are shown in Figure 9. Here it can be seen that again the
experimental response is quite linear (R2 = 0.9985), although note that the load levels are
higher and the displacements lower than the core only test, indicating a significant increase in
stiffness with the addition of the skin. For this configuration, the EB analysis code and the
SolidWorks Simulation FEA predictions both match the experiment quite well. In this case
there is a reduction in the experimental stiffness for tip displacements above roughly 12 mm,
such that the analytical model changing from underprediction at low displacements to
overpredicting at higher displacements. The FEA results underpredict the load required over
this displacement range, but the overall stiffness trend is better matched.
Figure 9. FishBAC stiffness under tip load, skin and core
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4.2 Stiffness under tendon moment
The results for the tendon moment driven, core only configuration are shown in Figure 10.
Once again, we see a very linear response from the experiment even though the loading
mechanism has changed. Note that the EB analysis code overpredicts the stiffness by a small
margin, and that the FEA overpredicts slightly more. Both show generally good agreement.
Figure 10. FishBAC stiffness under tendon moment, core only
The EMC skin was bonded on and the FishBAC re-tested under tendon moment, and,
as in the case of the tip load experiments, there was an increase in stiffness. Figure 11
compares the experimentally measured moment versus tip displacement results to predictions
from the EB analysis code and the SolidWorks FEA. In this case, both methods are
underestimating the stiffness, with the EM analysis providing a quite good prediction and the
FEA having larger disparities. It should be noted however, that the results of this experiment
are less linear than the others. This may indeed be the true behaviour of the FishBAC
structure under such loading, but it is also possible that some shifting of the clamping
mechanism or load application mechanism, or some other non-linearity, may have caused the
apparent shift in slope that occurs near 5 mm of tip displacement. The FEA predictions seem
to match the slope of the second portion of the test better than the EB analysis, and so the
accuracy of this method may be under represented in this case due to the specifics of the
experiment. It is hard to make a clear conclusion though from a single test, except to say that
both methods generally match well with the experiment.
Figure 11. FishBAC stiffness under tendon moment, skin and core
The results shown above are further summarized in Table 3, which compares the
stiffness values which result from linear regressions to the various experimental and predicted
responses. In this way the ability of the two analysis methods to obtain the desired levels of
14
accuracy can be compared. Here it can be seen that the predictive capability of the FEA for
the tip load cases is very good, however it is less so for the tendon moment cases. The EB
analysis code, however, is able to achieve the required error levels of less than 10% in all
cases, and does particularly well for the tendon moment with skin and core (error = 1.27%),
which is the most representative of actual implementation of the FishBAC concept, and the
most relevant for predicting the actuation requirements of the system.
Table 3. Summary of stiffness prediction results
Configuration Method Stiffness Units % Error
Tip load, core only
Experiment 0.1601 N/mm ─
EB Analytical 0.1508 N/mm 5.81
FEA 0.1648 N/mm 1.03
Tip load, skin and core
Experiment 0.3566 N/mm ─
EB Analytical 0.3726 N/mm 4.48
FEA 0.3532 N/mm 0.95
Moment, core only
Experiment 0.0139 Nm/mm ─
EB Analytical 0.0150 Nm/mm 8.06
FEA 0.0157 Nm/mm 12.95
Moment, skin and core
Experiment 0.0422 Nm/mm ─
EB Analytical 0.0417 Nm/mm 1.27
FEA 0.0380 Nm/mm 9.95
5 CONCLUDING REMARKS
In summary, this work has presented two different structural analysis methods for the Fish
Bone Active Camber concept and compared them to experimental measurements. Four test
cases were created with either a tip load or an internal tendon moment applied to either the
FishBAC core structure, or the FishBAC core with skin bonded on. The high fidelity method
was SolidWorks Simulation 2013, a commercially available finite element analysis software
which was found to provide good accuracy, particularly for the tip load cases. A low-fidelity
analysis was also shown which was derived from Euler-Bernoulli beam theory and which also
produced predictions with acceptable error levels. The much faster running time of the beam
theory analysis (seconds instead of minutes/hours) makes it more useful for the fluid-structure
interaction analysis and geometry/material optimization work which is currently ongoing.
ACKNOWLEDGEMENTS
The research leading to these results has received funding from the European Research
Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC
Grant Agreement n. [247045].
REFERENCES
[1] J.E. Herencia, P.M. Weaver and MI Friswell, “Morphing Wing Design via Aeroelastic
Tailoring,” 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and
Materials Conference,Waikiki, Hawaii, 23-26 April 2007, paper AIAA-2007-2217.
[2] O. Bilgen, E.I. Saavedra Flores, and M.I. Friswell, “Optimization of Surface-Actuated
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