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Fluid/Structure-Interaction Analysis of the Fish-Bone-Active-Camber Morphing Concept
Benjamin K. S. Woods, Iman Dayyani, and Michael I. Friswell
Swansea University, Swansea, Wales SA2 8PP, United Kingdom
A coupled, partitioned fluid/structure-interaction analysis is introduced for calculation of the deformed
equilibrium shape, aerodynamic coefficients, and actuation requirements of the fish-bone-active-camber morphing
concept. The fish-bone-active-camber concept is a high-authority morphing camber architecture with a broad range
of applications, including fixed-wing aircraft, helicopters, wind turbines, and tidal-stream turbines. The low
chordwise bending stiffness of themorphing structure, high stiffness of the tendondrive system, and the large changes
in aerodynamic loading while morphing necessitate a coupled fluid/structure-interaction analysis for determination
of the static equilibrium. An EulerBernoulli beam-theory-based analytical model of the structure is introduced and
validated.Aerodynamic loads are foundusing theXFOIL software,which couples a potential-flowpanelmethodwith
a viscous boundary-layer solver. Finally, the tendons are modeled as linear stiffness elements whose internal strains
are found from the EulerBernoulli theory, and whose axial forces create bending moments on the spine at their
discrete mounting points. Convergence of the fluid/structure-interaction code is stabilized through incorporation of
relaxation parameters. The results for two chosen test cases are presented to give an insight into the mechanical and
aerodynamic behavior of the fish-bone-active-camber concept.
A = cross-sectional areab = spancd = drag coefficientcl = lift coefficientd = diameterE = elastic modulusEI = flexural rigidityF = forceI = second moment of areak = axial stiffnessM = bending momentMa = Mach numberp = net aerodynamic pressureR = spooling-pulley radiusRe = Reynolds numberr = distance from the neutral axist = thicknessV = shear forceV = freestream velocityw = bending deflectionwA = Aitken relaxation parameterwF = fixed relaxation parameterx = normalized chordwise positionyten = tendon mounting offset = angle of attackcl = change in lift coefficientl = change in lengthF = spooling-pulley rotation = bending slope
bs = bending spineE = end of morphing sectioni = current iterationls = lower skinS = start of morphing sectionsk = skinst = stringerten = tendontot = totalus = upper skinw = due to bending = due to pulley rotation0 = initial
T HE development of compliance-based approaches formorphing-aircraft structures requires careful consideration ofthe methods used to model the aerodynamic and structural per-formance. Typically, if the deflections achieved with the morphingstructure are large, then the coupling between the aerodynamic andstructural loads will be strong. For this reason, a fluid/structure-interaction (FSI) analysis is crucial to the successful predictionof performance for these concepts. This work focuses on thedevelopment of an FSI code for a newly developed camber morphingairfoil known as the fish-bone-active-camber (FishBAC) concept. Anoverview of previous compliance-based camber morphing conceptswill be provided, along with a review of FSI methods applied tocompliant morphing concepts. After this, the design philosophy ofthe FishBAC concept will be briefly presented. The two partitions ofthe FSI code, the aerodynamic and structural solvers, will then bedetailed, and the coupling of the two will be discussed, with specialattention paid to the relaxation parameters employed to ensure thestability and convergence of the algorithm. Finally, the results fromthe FSI code will be shown for two representative design cases.
Asignificant amount of research effort has been put into the pursuitof compliant morphing camber mechanisms for both fixed- androtating-wing applications. The many different concepts explored
Presented as Paper 2013-1908 at the 54th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, andMaterials and Co-Located Conferences,Boston, MA, 2326 April 2013; received 14 November 2013; revisionreceived 4 August 2014; accepted for publication 6 August 2014; publishedonline 18 November 2014. Copyright 2014 by Benjamin King SuttonWoods. Published by the American Institute of Aeronautics and Astronautics,Inc., with permission. Copies of this paper may be made for personal orinternal use, on condition that the copier pay the $10.00 per-copy fee to theCopyright Clearance Center, Inc., 222RosewoodDrive, Danvers,MA01923;include the code 1542-3868/14 and $10.00 in correspondence with the CCC.
*Research Officer, College of Engineering. Member AIAA.Research Assistant, College of Engineering.Professor, College of Engineering.
JOURNAL OF AIRCRAFTVol. 52, No. 1, JanuaryFebruary 2015
over the last several decades have been well summarized in severalreview papers .Much of the work done to date has been focused on the
development of active rotor systems for helicopters and tilt rotors.The two most successful active rotor concepts to date have beendeveloped to the point of full-scale wind-tunnel testing and flighttesting. Straub et al. have developed a hinged trailing-edge flap drivenby piezoelectric stacks with mechanical stroke amplification .While this design is more akin to a traditional trailing-edge flap(albeit with smart-material actuation) than to a continuous cambermorph, it is relevant here as an example of the state of the art, becausean extensive open- and closed-loop vibration-reduction testing wasperformed over a broad range of operating conditions in the NASA40 80 ft wind tunnel. Eurocopter has flight tested a similar piezo-driven trailing-edge flap system on a highly instrumented BK117 .EADS, the parent company of Eurocopter, has also developed acontinuous active-camber morph driven by piezoelectric benders .While these designs have shown a significant potential to reducevibration, they do so with fairly modest camber deflections andincremental changes to the local airfoil properties. These concepts areboth intended for vibration reduction, and the effective flapdeflections obtained are on the order of several degrees. A morphingrotor system with a sufficient control authority to provide primarycontrol or to allow for significant reconfiguring of rotor propertiesremains an active area of research.Other concepts are being pursued for fixed-wing applications that
are intended to create larger changes in airfoil properties. Barbarinoet al. provide a thorough overview of work done to date . Ofparticular interest to this discussion are concepts that create smooth,continuous camber change. The Defense Advanced ResearchProjects Agencys Smart Wing program included a complianttrailing-edge morphing structure . This structure consisted of acentral laminate with honeycomb on top and bottom to support aflexible silicone skin. This concept was actuated by an eccentuator,which employs a rotating kinked rod that transforms the rotarymotion of a piezoelectric motor into thicknesswise translation ofseveral points along the chord of the morphing section, therebycreating a smooth bending deflection. A gear box was used toincrease the torque output of themotors, at the expense of bandwidth.Wind-tunnel testing on an unmanned-aerial-vehicle (UAV) modelfitted with a morphing trailing edge showed large deflections underaerodynamic loads up to Ma 0.8. While changes to sectionproperties are not presented, the total vehicle control moments areshown, and the concept was able to provide a reasonable controlauthority to the vehicle.A related concept is under investigation for wind-turbine
applications by Daynes and Weaver . Here, the laminate structureis moved out of the center of the section to become the upper-skinsurface, which is connected to the lower silicone skin surface througha Nomex honeycomb core. Actuation is provided by a gear motoracting through a rigid pushpull rod connected to the trailing edge.The undesirable anticlastic-curvature effects caused by the largePoissons ratio of the core are reduced by making a series ofchordwise cuts through the core every 20 mm. Without these cuts,honeycombs form a saddle shape when bent, creating a curvaturealong the span as well as along the chord. Wind-tunnel testing of thisconcept at speeds up to 56 ms showed a good control authority, withchanges to the local lift coefficient on the order of cl 0.5 for a20% chord morphing flap .Concepts that employ compliant structures to achieve camber
change will invariably need to consider the effects of aerodynamicloading on the deformed shape of the structure. This stems from thefact that designing a structure to be deflected with reasonableactuation loads generally leads to a level of compliance thatwill allowfor aerodynamic loading to have a nonnegligible impact on deformedshape. This therefore requires the use of coupled FSI solvers toproperly account for aerodynamic induced deflections to solve forequilibrium aeroelastic deflections. Bae et al. show the importance ofaerodynamic induced deformations for a morphing UAV usingpiezoelectric actuators to induce camber change . Campanileand Anders studied aerodynamic deformations and aeroelastic
amplification in the belt-rib concept proposed by DLR, GermanAerospace Center . In Barbarino et al., the effect of aerodynamicloading on a shape-memory-alloy-driven active-camber concept ispresented, but the aerodynamics and structures are not directlycoupled . Bilgen et al. present a coupled, partitioned FSI solverand optimizer for a thin shell morphing airfoil driven by skin-mounted macrofiber composite actuators . De Gaspari and Riccipresent a two-level optimization routine for morphing camber designusing FSI and genetic algorithms . Daynes and Weaver show astaged FSI analysis of a morphing wind-turbine blade, wherein thestatic solution is first foundwithout aerodynamic loading, afterwhichthe aerodynamic load was applied and the code run again untilconvergence . Molinari et al.  and Thuwis et al.  alsopresent related FSI analyses for morphing-aircraft applications.
III. FishBAC Concept
The FishBACmechanism provides an alternative design architectureto the compliant morphing concepts discussed previously. Introducedby Woods and Friswell , this design employs a biologicallyinspired compliant structure to create large, continuous changes inairfoil camber and section aerodynamic properties. The structure,shown schematically in Fig. 1, consists of a thin chordwise bendingbeam spine with stringers branching off to connect it to a pretensionedelastomeric matrix composite (EMC) skin surface. Both core and skinare designed to exhibit near-zero Poissons ratio in the spanwisedirection. Pretensioning the skin significantly increases the out-of-plane stiffness and eliminates buckling when morphing. Smooth,continuous bending deflections are driven by a high-stiffnessantagonistic tendon system. Actuators mounted in the nonmorphingleading-edge portion of the airfoil (which constitutes the primary load-bearing wing spar) drive a tendon spooling pulley through anonbackdrivable mechanism (such as a low-lead-angle worm andworm gear). Rotation of the pulley creates equal but oppositedeflections of the tendons. These differential displacements generate abending moment on the rigid trailing-edge strip, thereby inducingbending of the trailing-edgemorphing structure to create large changesin airfoil camber. Because the tendon system is nonbackdrivable,no actuation energy is required to hold the deflected position ofthe structure, reducing control action and power requirements.Furthermore, the automatic locking action of the nonbackdrivablemechanism allows the stiffness of the tendons to contribute to thechordwise bending stiffness of the trailing edge under aerodynamicload, without increasing the amount of energy required to deflect thestructure.The wind-tunnel testing of the prototype seen in Fig. 2 found that
the FishBAC provided improved aerodynamic efficiency comparedto traditional trailing-edge flaps, with increases in lift-to-drag ratio of2025% being realized at equivalent lift conditions . An increasein the lift coefficient of cl 0.72 between unmorphed andmorphedwasmeasured at a freestreamvelocity ofV 20 ms andan angle of attack of 0 deg, and this cl was maintained overthe entire nonstalled range of alphas. The freestream velocity ofV 20 ms, while perhaps low for many of the potentialapplications, was chosen chiefly due to limitations in thewind-tunnelinstrumentation.Testing to date has focused on quasi-static and low-frequency
deflections of the FishBAC structure, on the order of a couple hertz.The primary limit on higher-bandwidth operation will be theactuation technology used. For this and other reasons, the FishBAC isnot tied to any particular actuation system. Any means capable ofgenerating rotations and torque that can be made to fit within the
Fig. 1 FishBAC concept.
308 WOODS, DAYYANI, AND FRISWELL
geometric constraints of the airfoil can be considered. Obviously,higher-frequency applications, such as vibration reduction on activehelicopter rotors (whichwould require actuation at 40Hz and above),will be more demanding than low-frequency ones due to the higherpower requirements. Hysteretic losses in the elastomer skin will alsoincrease with frequency. The actuator independence built into theconcept is therefore intended to allow the actuation to be scaled tobest meet a particular set of requirements. Work is currentlyunderway into the design and testing of a high-bandwidth FishBACdemonstrator using advanced high-power brushless motors andcompact high-efficiency gearing strategies.The large achievable deflections and continuous compliant
architecture make this concept potentially applicable to fixed-wingapplications ranging in scale from small UAVs to commercialairliners, and to rotary-wing applications including wind turbines,helicopters, tilt rotors, and tidal-stream turbines.
IV. FSI Analysis
The FSI analysis developed for the FishBAC structure uses twopartitioned codes to separately analyze the aerodynamics andstructural mechanics, with the antagonistic tendon-actuation systembeing incorporated into the structural mechanics. These codes arecoupled and iterated until convergence is achieved for all the relevantaerodynamic coefficients, as shown schematically in Fig. 3. Themismatch in stiffness between the stiff tendon drive system and theaerodynamic load requires the use of a relaxation parameter toensure stability of convergence. Two different relaxation-parameteralgorithmswere investigated in this work, and evaluated based on thespeed and stability of their convergence.
A. Airfoil Definition and Deflection Parameterization
To ensure compatibility between the aerodynamic and structuralsolvers, the parameterization of the airfoil shape must be consistentfor both. This was accomplished by independently defining thethickness distribution and the neutral-axis deflection. In this way, thedefinition of the skin surface geometry required for the aerodynamicsolver could be found for any deflected shape by superimposing theknown thickness distribution onto the neutral-axis position. Asymmetric NACA 0012 airfoil was used as the baseline for theFishBAC design shown here, and the internal compliant core,tendons, and elastomeric skin were also symmetric in their geometryand properties. Because of this, the neutral axis lies at the center of theairfoil, following the chord line of the undeformed state. Thepresence of the stringers in the FishBAC design serves to enforce theoriginal thickness distribution in the skin, even with large trailing-edge deflections. While the skin does experience out-of-planedeflections between the stringers, they are of small magnitude, andtheir effect on the overall bending stiffness of the FishBAC can beignored in this low-fidelity analysis. The effect of these local skindeflections on aerodynamic performance is perhaps not negligible,however, aswill be discussed in further detail in Sec. IV.B.Because of
the structural symmetry and stringer enforced thickness distribution,the FishBAC airfoil can be parameterized as a constant thicknessdistribution overlaid onto the deflected shape of the bending beamspine. It is important, however, to account for curvature effects byadding the thickness distribution in a direction that is locally normalto the bending spine, especially given the large amount of camberobtainable with this concept. This approach is equivalent to themethodology used to define cambered airfoils for the NACA four-digit series airfoils .Only the portion of the airfoil behind the nonmorphing spar is
considered in the structural model of the FishBAC, but the entireouter-skin surface is needed for the aerodynamic analysis. Themorphing section begins at the nondimensional chord location, xS,and the morphing section continues until the beginning of the rigidtrailing-edge strip, xE, as labeled in Fig. 1. It is important to note that,while the rigid leading edge is not included in the structural model ofthe FishBAC, the rigid trailing-edge strip is. This is because, eventhough it is assumed rigid and does not undergo any strains of its own,it does carry aerodynamic-pressure loads that must be resolved asshear forces and bending moments through the spine.
B. Aerodynamic Model
The aerodynamic-pressure distribution acting on the FishBAC isfound using the XFOIL panel-method code . This code is basedon potential-flow theory with the addition of a viscous boundary-layer solver to predict skin-friction drag and flow separation, offeringa more complete drag prediction than inviscid codes. XFOIL hasbeen shown to correlate well with high-fidelity computational fluiddynamics simulations for the types of smoothly cambered deflectedshapes typical of the FishBAC, over a considerable range ofmorphing start locations, amounts of camber, angles of attack, andReynolds numbers . It is also computationally inexpensive andeasy to integrate intoMATLAB, and so provides an ideal solution forthe low-fidelity analysis performed here. XFOIL requires as inputsthe aerodynamic conditions (Mach number Ma, angle of attack ,and Reynolds number Re) and the nondimensionalized airfoil skincoordinates. Using a viscous formulation of linear vorticity potential-flow theory, XFOIL then calculates the distribution of pressurecoefficient over the airfoil. Figure 4 shows a typical result of thepressure-coefficient distribution over the upper and lower surfaces ofthe entire airfoil. The airfoil geometry and operating conditions forthe result shown are outlined in Tables 1 and 2, and the spooling-pulley rotation is set to F 50 deg. The lift, drag, and momentcoefficients for the current airfoil shape are found by XFOIL fromthis pressure-coefficient distribution with inclusion of the viscousboundary-layer effects.For the structural component of this FSI problem, we consider
the distribution of the dimensional pressure acting only on themorphing portion of the chord (xS < x < xE). Furthermore, because
Fig. 2 FishBACwind-tunnel-test model showing baseline andmorphedshapes.
Fig. 3 FSI algorithm schematic.
WOODS, DAYYANI, AND FRISWELL 309
deformations in the thickness direction are not considered in thisanalysis (because the thickness distribution is assumed to bemaintained by the stringers), the pressure on the lower and upperskins can be combined to create a net pressure that acts on the spine asa chordwise varying distributed load, p, as shown in Fig. 5. Theresults shown here are for the same configuration used in Fig. 4. Forthis analysis, the net-pressure distribution is applied directly to thebending spine. While in reality the load application would be morediscretized, with each stringer carrying a locally integrated skinpressure load into its discrete attachment point with the spine, thenumber of stringers is fairly large and the changes in pressure aregradual enough that applying a continuous pressure load is areasonable simplifying assumption.The local out-of-plane deformations of the skin in between
stringers are not directly considered in this FSI analysis. This isdriven by the fact that the aerodynamic code used does not have high-enough fidelity to accurately resolve the changes in aerodynamicpressure created by such deflections. While this is a limitation thatcould only be completely overcome by a higher-fidelity (andtherefore, more computationally expensive) aerodynamic compo-nent, the impact of this assumption is mitigated by two factors. First,the pretensioning employed in the elastomeric skin of the FishBACconcept significantly reduces the magnitude of any out-of-planedeflection, and second, the aerodynamic pressure acting on the
morphing region of the FishBAC is much smaller than that seen onthe rigid leading-edge spar, as can be seen in Fig. 4.
C. Structural Model
The derivation of the analytical formulation for morphing-sectionstiffness from EulerBernoulli (EB) beam theory will be shown, theboundary conditions will be presented, and the distribution offlexural rigidity along the chord will be formulated. An overview ofthe structural-modeling approach detailed as follows is shownschematically in Fig. 6.
1. Governing Equations
The structural model used in this FSI analysis is analytical andderived from the EB beam theory. The low bending stiffness and highlength-to-thickness ratio of the bending spine, low in-plane stiffnessof the skin, and the continuous loading along the span make the EBtheory a good initial approximation for analysis. Furthermore, thehigh-stiffness stringers branching off from the spine help to enforcethe plane sections remain plane and normal assumption, whichunderlies the derivation of the EB beam theory . Furthermore,empirical evidence, such as Fig. 2, indicates that bendingdeformations do indeed dominate over shear deformations, as seenby the normal orientation of the stringers relative to the spine.Ultimately, of course, the accuracy of the model must be validatedagainst other trusted methods, as will be done in Sec. V.The bending deflections of the morphing structure are found by
integrating the aerodynamic-pressure distribution to find the sheardistribution, integrating that to find the moment distribution,integrating again to find the distribution of slope, and then integratingone final time to find the deflection of the neutral axis. The effect ofthe actuation moments from the tendon system is added as additionalbending moment during this integration process.The relationship between net aerodynamic pressure p, flexural
rigidity EI, and vertical displacement w is given by the EB beamequation :
Table 1 FishBAC prototype parameters
Baseline airfoil NACA 0012Chord c 305 mmSpan b 150 mmStart of morph xS 0.35c 107 mmEnd of morph xE 0.85c 260 mmSpine thickness tbs 2 mmNumber of stringer pairs 14Stringer thickness tst 0.8 mmSkin thickness tsk 1.5 mmTendon offset yten 4.2 mmTendon diameter dten 0.7 mmSpine modulus Ebs 2.14 GPaStringer modulus Est 2.14 GPaTendon modulus Eten 131 GPaSkin modulus Esk 4.56 MPa
Table 2 FishBAC operatingpoint for FSI test case
Freestream velocity V 20 msReynolds number Re 240,000Angle of attack 5 degTendon pulley angle F 050 deg
Fig. 5 Representative net-pressure distribution (F 50 deg).
Fig. 6 Schematic of the structural-modeling approach.
Fig. 4 Representative distribution of pressure coefficient
(F 50 deg).
310 WOODS, DAYYANI, AND FRISWELL
Integrating pressure produces force, specifically the vertical shearforce V acting on the beam:
Zpx dx Vx (2)
The integral of shear force is bending momentM:
dx2ZVx dx Mx (3)
The curvature of the spine can be found by rearranging Eq. (3):
dx2 MxEIx (4)
Curvature is then integrated to give slope :
dxZMxEIx dx x (5)
And finally, integrating slope provides the distribution of verticaldeflection:
x dx (6)
The integration constants that are generated during the solution of theEB equations are solved by considering the boundary conditions. Forthe FishBAC, the bending spine is assumed to be clamped at itsattachment to the nonmorphing spar, and the end of the trailing edgeis assumed to be free. This leads to the following conditions:
wxr 0 (7)
xr 0 (8)
M1 0 (9)
V1 0 (10)
2. Flexural-Rigidity Formulation
The flexural-rigidity EI distribution of the FishBAC structure isformulated as a linear superposition of its components in thisanalysis. The spine is modeled as a constant-thickness beam, andbecause its neutral axis is coincident with that of the FishBACstructure as a whole, the flexural rigidity of the bending spine EIbs isequal to
EIbs Ebs12bt3bs (11)
in whichEbs is the elastic modulus of the bending-spinematerial, b isthe span of the FishBAC segment, and tbs is the thickness of thebending spine. Note that the rigidity of the spine could be madeto vary along the chord without changing the formulation orimplementation of this analysis. Indeed, tapering the thickness ormodifying the material properties of the spine is an effective meansof controlling the deflected shape, and therefore, aerodynamicproperties, of the FishBAC.While the results presented here are for aconstant-thickness spine, current work by the authors seeks to ex-ploit these design variables to optimize the structure for variousperformance 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 stiffnessmust be considered. Theflexural rigidity of the skin on the FishBACEIsk is thereforemodeledusing the parallel-axis theorem . The upper and lower skins areconsidered separately, and then added:
Here, Esk is the elastic modulus of the skin material, tbs is thethickness of the skin, rls is the distance between the lower-skinsurface and the neutral axis, and rus is the equivalent distance for theupper-skin surface. As with the spine, the skin, in this initial analysis,is a constant thickness along the chord. However, in this case, theflexural rigidity is not constant along the chord due to the changingthickness of the airfoil. Significant reductions in stiffness, andtherefore, actuation requirements, could likely be obtained with nodetrimental effect on the maximum out-of-plane displacement if theskin thickness were tapered to match the distribution of theaerodynamic-pressure coefficient.The stringers are included into the structural analysis in a similar
manner. For the portions of the spinewhere they exist, they contributesignificantly to the flexural rigidity due to their large thickness.Again, their flexural rigidity is found in the standard way:
EIst Est12bt3st (13)
The total FishBAC flexural rigidity EItot is then found as the linearsum of the spine, skin, and stringer rigidities:
EItot EIbs EIsk EIst (14)
Note that, due to the discontinuities in flexural rigidity caused by thestringers, this summation is done numerically in vector format, withthe flexural rigidity for a series of points along the chord beingcalculated. These points are chosen in a manner that captures thesudden changes in rigidity present by placing a point on either side ofthe discontinuity with a very small offset. This is important tominimize errors caused during the numerical integration of the EBbeam equations. A typical distribution of flexural rigidity over themorphing section is shown in Fig. 7. Note that the stringer EI isseveral orders of magnitude larger than that of the skin and bendingspine. For this reason, a zoomed-in inset plot has been included. It canbe seen that the contribution from the bending spine is constant alongthe chord (due to constant thickness), whereas that of the skindecreases due to the reduction in airfoil thickness toward thetrailing edge.
D. Antagonistic Tendon-Actuation Model
The effect of the tendonmoment on the structure is incorporated byadding an additional moment to the calculated bendingmoment fromaerodynamic loading. The tendon bending moment is applied at theanchor point for the fixed ends of the tendons, which coincides with
Fig. 7 Chordwise distribution of flexural rigidity.
WOODS, DAYYANI, AND FRISWELL 311
the beginning of the rigid trailing edge, xE. The tendon momentresults from strains induced by the prescribed spooling-pulleyrotation angle F and the FishBAC bending induced strains. Thetendon is mountedwith a sizeable offset from the spine neutral axis toallow it to effectively induce bending moments, which then leads tosignificant changes in length in the tendon with FishBAC deflection.Changes in tendon length change the force levels in the tendon andthe resulting bending moments applied to the trailing edge. Themoment from the tendons, aerodynamic loading, and the resultingFishBAC deformations are therefore highly coupled. To include thetendons in the FSImodel, wemust first calculate the kinematics of thevariation in tendon length with actuation inputs and FishBACdeflection, and then formulate the stiffness of the tendons to relatedeflections to resulting applied moments. Figure 8 showsschematically the forces and moments generated by the tendonsdue to pulley rotation, the magnitudes of which will now be derived.The total change in tendon length is a combination of the change
in length due to pulley rotation l and that due to deflection in thespine lw:
ltot l lw (15)
The change in length due to pulley rotation is found from the arc-length formula from the spooling-pulley radius R and the spooling-pulley rotation angle F:
l RF (16)
Additionally, deformations of the spine induce changes in length ofthe tendons because they are supported at a distance from the neutralaxis. In the current design, the distance from the neutral axis is fixedand equal to the tendon offset at the trailing-edge mounting point,yten. This is done by running the tendons through small orifices in thestringers at a constant distance from the neutral axis.While the tendonis free to move closer or further away from the neutral axis in theunsupported regions between stringers, it is assumed in this analysisthat the stringer spacing is close enough that these deviations will bequite small in magnitude, and the changes in overall tendondeflection caused by them will be correspondingly small. Given thisassumption, changes in tendon length due to spine bending can thenbe found by integrating the product of the bending-spine curvatureand the distance from the tendon to the neutral axis according to theEB theory :
dx2yten dx (17)
The stiffness of the tendon is found from the linear elastic theory foran axial tension rod. The sliding motion between the tendons and thestringers is assumed to occur without friction, and the use of braidedcord for the tendons gives them negligible inherent bending stiffness.The tendon stiffness kten is therefore equal to
in which Lten is the original tendon length, and the cross-sectionalarea Aten is found from the diameter:
The force in the tendon at a given deflection will then be
Ften ktenltot (20)
And finally, the total bending moment applied to the trailing edge byboth tendons can be found from
Mten 2Ftenyten (21)
Note that both tendons contribute equally to the generation ofmoment, such that the total moment is twice that of a single tendon.While this may seem counterintuitive, this result can be explained asfollows. Because of their equivalent distance from the neutral axis,both tendons in the antagonistic pair will experience the samemagnitude of length change from pulley rotation and spine bending;however, the lower and upper tendons will have different directionsof length change. Specifically, for the case of a downward FishBACmorphing, the lower tendon will have a decreasing length due toinitial pulley spooling, and a decreasing length with increasing spinedeflection. The upper tendon, on the other hand, will be subject to anincrease in length due to initial pulley rotation and increasing lengthdue to increased spine deflection. Countering the opposite signs ofthe upper and lower tendon deflections is the opposite sign of thebending moment created by each. A positive pulley rotation(clockwise) will produce an increasing tension force in the lowertendon, increasing the positive (flap down) moment it induces on theFishBAC. Similarly, a positive pulley rotation reduces the tensionforce on the upper tendon, decreasing the negative (flap up) momentit induces on the FishBAC. In this way, the stiffnesses of both tendonscontribute to themoment applied to the trailing edge. The tendons canonly carry tension, however (due to their lack of compressivestiffness), and so if the elastic deflections in the upper tendon and thebending deflections in the spine are such that the total tendonextension is less than zero, it will no longer contribute to the systemstiffness. Pretensioning the tendons sufficiently ensures this does nothappen over the range of deflections required. The pretension forcesare balanced, and so produce no net moment on the spine.Additionally, as long as the tendons do not leave their linear elasticrange, pretensioning has no effect on tendon axial stiffness.
E. Relaxation Parameters
FSI algorithms frequently include relaxation parameters to reducethe occurrence of large divergent oscillations in the predicteddisplacements between iterations . Relaxation parameters work byadding numerical damping to the solution,whereby the solution is onlypartially moved toward the solution predicted for the next iteration. Inthis way, the change in forcing experienced is reduced, and thetendency to experience fluctuating solutions of increasing divergenceis tempered. If properly formulated, relaxation parameters do notchange the value of the final converged solution, although they candecrease the speedof convergence.Therefore, they canbe thought of astrading computation speed for solution stability. They are particularlyimportant in systems with large differences in the stiffness of thevarious components. In the case of the FishBAC, the tendons aresignificantly stiffer than the spine, which is stiffer than theaerodynamics, making relaxation essential for stability. Two differentrelaxation-parameter formulations were considered in this work,namely, fixed and Aitken relaxation parameters. While relaxationparameters are typically applied to the displacement boundary betweenthe structure and the fluid , itwill be shown that convergence in thisproblem was considerably improved by also including a relaxationparameter on the interaction between the tendon-actuation system andthe structure.
1. Fixed Relaxation Parameter
A simple form of damping can be added to the solution process byusing a fixed relaxation approach, which allows for adjustableweighting of the solution toward the previous result :
ui1 wF ~ui1 1 wFui (22)
Fig. 8 Schematic representation of the antagonistic tendon actuation.
312 WOODS, DAYYANI, AND FRISWELL
in which the fixed relaxation parameter wF is a constant value tunedfor the given analysis, and ~ui1 is the estimated solution (be it spinedisplacement or tendonmoment) for the next iteration. Avalue ofwFclose to zero will lead to a very slow but stable convergence, whereasvalues close to 1 will essentially remove the relaxation effect,increasing the risk of instability.
2. Aitken Relaxation Parameter
More sophisticated algorithms can often improve the speed ofconvergence without affecting stability by using adaptive relaxationparameters. A commonly used adaptive method is the Aitkenrelaxation parameter . This method is computationally cheap, asit is calculated from the results of the two previous runs of the code,and therefore, does not require additional FSI runs . In thisalgorithm, the amount of difference between estimates, known as theresidual, is used to predict the value of the Aitken relaxationparameterwA, whichwill produce a residual of zero. The relaxation istherefore adaptive to the conditions of the solution, which oftenresults in improved convergence. The general form of the solutionusing Aitken relaxation is
ui2 ui1 wA;i1 ~ui2 ui1 (23)
Note that theAitken relaxation parameterwA is itself recursive, and isdefined from
wA;i1 wA;iui ~ui1
ui ~ui1 ui1 ~ui2(24)
Again, the parameters with tildes are estimated values (i.e., the rawoutput from the FSI before relaxation is applied).
3. Application to FishBAC FSI
Typically, relaxation parameters in FSI problems are applied to thesolution of the displacement of the boundary between the fluid andthe structure. Because the analysis here essentially has three coupledcomponents (fluid, structure, and actuation tendons), applying arelaxation parameter to both the fluidstructure interface and thetendonstructure interface was found to significantly improveconvergence. Specifically, the bending-spine deflectionwx and thetendon momentMten were relaxed. The need for this can be seen inFig. 9, which shows a convergence plot for lift coefficient for thegeometry given in Table 1 and the design case given in Table 2, withF 50 deg. Three lines are plotted, which show the impact ofapplying a fixed relaxation parameter to various parts of the solution.The particular case shown here uses the FishBAC geometry outlinedin Table 1 at the operating point shown in Table 2 and a spooling-pulley rotation of F 50 deg. Note that, if the relaxation is appliedonly to the spine, which represents the fluid/structure-interfacedeflection, which is the typical approach of other FSI algorithms,then the solution quickly diverges. This is due to very largefluctuations in the tendon moment. Even small spine deflectionscause significant changes in tendon length, which, due to the highstiffness of the tendons, generates very-large-magnitude bendingmoments, which quickly drive the solution to diverge.It can be seen in Fig. 9, however, that applying a relaxation
parameter to just the tendon solution is sufficient to stabilize theconvergence of lift coefficient. Relaxing both the spine and thetendon moment produces the best result though, providing a quickand stable approach to the converged solution with minimalovershoot. For the results shown here, the relaxation parameters werechosen through a simple trial-and-error process, starting with thedesign case mentioned previously, which had a large deflection, andtherefore, significant coupling. A tendon-moment relaxation factorof wFtendon 16 and a spine-deflection relaxation factor ofwFspine 13 gave good performance. Further manual ex-perimentation showed that these values worked well for a wide rangeof configurations.Comparing the Aitken relaxation to the fixed relaxation gives
another interesting result. As can be seen in Fig. 10, the fixed
relaxation parameter is found to provide faster convergence. Again,the geometry used is that of Table 1, the operating point is given inTable 2, and the spooling-pulley rotation was F 50 deg. Whilethe particular values used for the fixed relaxation have been tuned(whereas the Aitken parameter is essentially self-tuning), the valueschosen work well over a range of angles of attack, freestreamvelocities, and actuator settings. For this reason, the fixed relaxationparameters given previously were used throughout this work,including all of the results to be presented as follows.
V. Validation of the Structural Model
To establish the accuracy of the EB beam theory structural modeldeveloped here, a series of experimental tests and finite elementanalysis (FEA) were carried out with different structural con-figurations and loading conditions.A FishBAC prototype developed in a previous work  was
tested with and without its EMC skin attached under tip loads andinternal tendon moments, creating four different configurations forvalidation of the structural model. While the geometry of the coreremained the same throughout the experiments, testing with andwithout the skin helps build confidence in the accuracy of the modelbecause it allows for an independent examination of the impact of theskin on overall stiffness. Because of the fact that it is a tensionedelastomeric skin that is only intermittently attached to the core, it isnot obvious that it would behave in a manner consistent with theassumptions of the EB beam theory, and therefore, it is useful to beable to isolate this particular aspect of the FishBAC structure.In addition to the EB model, a high-fidelity FEA of the structural
configurations was run in Abaqus/CAE 6.11. Two FishBAC modelswith and without elastomeric skin were generated in Abaqus. Beamelements with cubic formulation were used to discretize the FishBACmodels. Small linear deformation theory was considered in the FEA.In bothmodels, all degrees of freedom for the leading-edge end of themorphing section were fixed. Concentrated loads or moments werethen applied to the relevant locations on the trailing-edge end asrequired.The geometric properties of the prototype and Abaqus models are
shown in Table 1. Further details of the construction of the prototypeare available elsewhere . Generally speaking, this prototype has achord representative of amedium-sizeUAVwing or a small rotorcraftblade. Note that this prototype does not include the nonbackdrivablemechanism.The first round of testing characterized the bending stiffness of the
prototype under tip loading. This is the simplest form of loading thatcan be used, and provides a useful baseline test for comparison. Theapparatus used is shown in Fig. 11. The forward portion of themorphing section was rigidly clamped, and a known displacementwas applied near the tip of the trailing edge using a digital heightgauge. The forces at the loading point were then measured with astrain-gauge load cell with digital readout. The core was first testedwithout the skin, and then tested again after the skin had beenbonded on.In Fig. 12, the experimental results for this tip-load testing are
compared to predictions from the EB code model and the AbaqusFEA. Note that the results are grouped into skin off and skin on, andthat the stiffness contribution of the skin is significant. It can be seen
Fig. 9 Application of fixed relaxation parameter to various systemcomponents (F 50 deg).
WOODS, DAYYANI, AND FRISWELL 313
that the experimental response is linear despite tip deflections, whichare quite large (up to 20% of the cantilevered length), and that whilethe skin does add significantly to the stiffness of the structure, it doesnot make its response significantly less linear. This implies that overthe magnitudes of skin strains realized in this test, the generallyhighly nonlinear elasticity of the elastomer can be successfullylinearized. In addition, the EBmodel provides a good estimate of themeasured deflections with and without the skin. As expected, theFEA also predicts the response well.For the second set of stiffness tests, the structure was deformed by
internal tendon moments in a manner closely approximating theactuation method for this concept. In this experiment, the base of theFishBAC was clamped vertically in a vise. Tendon moments weregenerated on the trailing edge of the FishBAC by attaching aSpectra fiber tendon to the anchor point on the trailing edge, andthen threading it through orifices in the stringers until it left theFishBACcore at its leading-edge extent, simulating the kinematics ofthe tendon actuation. Masses were then suspended from the exposedportion of the tendon to generate a known tension in the tendon. Thistension is converted into amoment using the known tendonmountingoffset (yten 4.2 mm). Figure 13 shows the setup for this test; notethat the tendon is not quite visible behind the steel support structure.
The results of testing with and without the skin bonded on areshown in Fig. 14. Again, there is a large difference in stiffnessbetween the skin-on and skin-off cases. For the skin-off case, the EBmodel slightly overpredicts stiffness while the FEA provides anexcellent agreement. For the skin-on case, the measured behaviorshows some nonlinearity, perhaps attributable to friction effectsbetween the tendon and the stringer orifices through which it passes,which are not included in the EB model or FEA. Despite this, theagreement of both models is quite good.Thus, it can be seen that the EB-beam-theory-based structural
model developed here provides a useful level of prediction accuracyfor these structures, comparable to that achieved with a high-fidelityFEA code. It is not immediately obvious that a simple linear analysis,such as the EB beam theory, would be applicable for the largedeflections seen here. However, when the small thickness of thespine, which acts as the primary core of the structure, is considered,the strains experienced in this very long and slender beam will bequite small. While the strains in the skin are indeed larger, they arestill within the linear range of the highly compliant elastomermaterialused. For this reason, the simple linear analysis is capable ofaccurately predicting the response of the structure for the large levelsof deflection measured in the experiments. In addition to the resultsshown here, further validation of the structural model has beencarried out in other work by the authors using a different airfoilconfiguration, again with very good agreement between the modeland the experiment despite large displacements .If we also consider that the EB structural model was written in the
same MATLAB language as the rest of the FSI code (facilitatingintegration and geometry definition), and that it allows for a completeanalysis of a configuration significantly faster than Abaqus, it is clearthat the EB model is the more useful option in the context of thiscurrent work.
VI. Results and Discussion
This section will present the results generated by the FishBAC FSIanalysis for two different representative geometries and aerodynamicoperating points. Initially, the results of each step in the structural andaerodynamic solvers will be shown for a baseline case, followed bythe convergence histories of the aerodynamic parameters of interest.The results for different tendon spooling-pulley rotation angles Fwill be plotted to show the effect of actuation and the resultingmorphing. Finally, the impact of actuation will be shown directly, by
Fig. 10 Comparison of lift-coefficient convergence for fixed and Aitkenrelaxation parameters (F 50 deg).
Fig. 11 Tip-load experimental setup (shown with FishBAC skin on).
Fig. 12 Force vs tip displacement under tip load.
Fig. 13 Tendon-moment experimental setup (shown with FishBACskin off).
314 WOODS, DAYYANI, AND FRISWELL
plotting the increases in lift, drag, and tendonmoment that occur withtendon spooling-pulley rotation.The test case used for these results was chosen to match the
operating conditions of the Swansea University wind tunnel, suchthat the FSI code could be used to design experimental test models.The geometric and material parameters were equivalent to thosepresented inTable 1, and the operating pointwas that listed inTable 2.The aerodynamic loading produced by the range of tendon pulley
rotation angles studied is shown in Fig. 15. In Fig. 16a, the solid linesare for the lower surface of the airfoil and the dashed lines are for theupper surface. It can be seen that increasing the pulley rotation leadsto an increase in the pressure loading. The net-pressure results inFig. 15b show an interesting trend of a dip in pressure that movesinward from the trailing edge with increasing pulley rotation. Thislikely indicates the onset of trailing-edge flow separation, whichwould be expected to move toward the leading edge as the bendingdeflections grow and the adverse pressure gradient on the uppersurface becomes steeper.The FSI code integrates the net-pressure results from Fig. 15b to
produce the vertical shear-force distributions seen in Fig. 16a. Notethat the magnitude of the forces increases with pulley rotation,although there is less change with higher rotation angles due to theprogression of flow separation seen in Fig. 15b. Figure 16b shows thebending moment produced by the integration of the shear forces.Note that these results also include the tendon moment. The tendonmoment has the effect of shifting the bending moment down by aconstant amount over the range xS < x < xE, with the amount of shiftincreasing with pulley rotation angle. This is the expected resultbecause larger pulley rotations would be expected to generate largeractuation moments, and as will be seen, larger displacements.Integrating bending moment and accounting for the flexural-rigiditydistribution give the slope results shown in Fig. 16c. Note the steppednature of the slope results, which is caused by small regions ofnegligible slope change at the locations of the high-stiffness stringers.The effect of the rigid trailing-edge strip can also be seen as a regionof essentially constant slope for xE x 1. The final integration stepproduces the bending-spine displacements shown in Fig. 16d. As
expected, the displacement increases with pulley rotation angle.Also, despite the sharp discontinuity in the bending-momentdistribution caused by the location of the tendon-moment ap-plication, and the steps in the slope results, the final deflected shapeof the spine is smooth and continuous. The maximum displacementof wTE 55.4 mm occurs when F 50 deg, and is equal to18% of the chord.By overlaying the airfoil thickness distribution on the bending-
spine displacements, we can see the predicted shape of the morphedFishBAC airfoil. Figure 17 shows these profiles for the range oftendon pulley rotation angles studied, along with a schematicrepresentation of the spooling-pulley rotation drawn to scale.Figure 17 highlights one of the limitations inherent to the EBformulation used here, which is the neglecting of any chordwisemotion of the end of the trailing edge. Because of the fixed length ofthe bending spine, the trailing edge would, in reality, experience asmall amount of chordwise motion, moving to the left as it bendsdown. The effect of the simplified treatment of the trailing edge usedhere is to unrealistically increase the arc length of the spine as itdeforms down. The magnitude of this effect is fairly modest, witha 3.7% increase in spine arc length for F 50 deg, whichcorresponds to an increase in the overall airfoil camber-line length of2.4%. It remains to be seen, however, if this formulation will lead toerrors in the structural or aerodynamic analysis, particularly at largercamber deformations. Future work will investigate this further.It is useful to consider the impact of tendon spooling-pulley angle
on the convergence of the aerodynamic coefficients. Of primaryinterest are the lift and drag coefficients visible in Fig. 18. Here,several notable trends can be seen. First, in Fig. 18a, it can be seen thatincreasing the pulley rotation increases the lift coefficient. This is anexpected result stemming from the increased spine deflections seen inFig. 16d, which would increase the airfoil camber, and thus, the lift.There is a diminishing return, however, on the amount of additionallift developed with each 10 deg increment of pulley rotation. This islikely to be a direct result of the pressure loss resulting from the flowseparation discussed previously. Despite this, the predicted range ofachievable lift coefficients is quite large, with cl 1.21 at thechosen operating point. Also, note that the convergence behavior isvery stable across the entire F range, with convergence occurring infour to five iterations. The drag results present an interestingcounterpoint. The drag coefficient in Fig. 18b is seen to grow at anaccelerating rate with increasing pulley rotation. This is due to thelarge drag induced by the onset of flow separation. Furthermore,while the convergence is again stable across the entire range, thenumber of iterations required to get a converged result increases withpulley rotation, where before it was essentially independent of F.This indicates that drag is more sensitive to spine shape than lift.The effect of pulley rotation on lift and drag can be considered
more directly by looking at only the final converged results, and byrunning the FSI code for a finer sweep of rotation angles. Figure 19shows the results for lift and drag with F swept from 0 to 50 deg in2.5 deg increments.
Fig. 14 Tendon moment vs tip displacement.
Fig. 15 Aerodynamic loading for test case a) pressure coefficient and b) net pressure over morphing region.
WOODS, DAYYANI, AND FRISWELL 315
The trends mentioned previously can now be seen in more detail.Figure 19a shows the diminishing returns in lift achieved with pulleyrotation. The results do not seem to have plateaued, however,indicating that still higher lift coefficients are theoretically possible.The realistic impact of this fact is probablymoot, however, as XFOILis generally not conservative in its lift and drag estimates, and realflow conditions in awind tunnel or in free flight would likely lead to amore rapid and extensive flow separation, and therefore, lower liftcoefficients. Indeed, cl 1.8 is a very high figure to achieve with asingle-element airfoil with no boundary-layer control . Similarly,the drag coefficients predicted in Fig. 19b are likely to be lower thanthose experienced experimentally. These limitations in XFOIL arewell known and not a problem from a design standpoint, so long asthey are properly accounted for. Given the analytical nature of theresults presented here, an experimental validation is ultimatelyrequired, and work to this end is ongoing.Another useful aspect of this FSI code is its ability to predict
actuation requirements. By prescribing the tendon spooling-pulleyrotation angle (equivalent to the required actuator output rotation),and then solving for the resulting moments generated by the tendons,the reaction moment that the actuator is required to maintain isautomatically known. Figure 20a shows the relationship between thepulley rotation and the required moment, and Fig. 20b shows theevolution of the predicted lift coefficient with the actuation moment.From a design standpoint, this is a very useful feature of the code,
because it allows for the actuation system to be sized appropriately.Figure 20a also includes the tendon-moment results for the samegeometry configuration, but with no applied aerodynamic loading.This provides insight into the relative stiffness of the structural andaerodynamic loads. For the fairly low-speed condition studied here,the aerodynamic pressures on the trailing edge are quite low, and somost of the torque required to deflect the FishBAC is from thestructural stiffness.We will now consider a second set of results corresponding to a
higher freestream velocity, one which is perhaps more representativeof the types of real-world applications envisioned for the FishBAC.As is the casewith traditional aircraft structural design, the design of aFishBAC morphing camber airfoil is strongly dependent on themaximum loads it will experience. Increasing operating velocitieswill increase the dynamic pressure acting on the compliant structure,necessitating a stiffer baseline structure and increased actuationrequirements. If we consider a M 0.3 case (V 102 ms) asgenerally representative of a medium-scale UAV, a large-scale windturbine, or the retreating blade of a rotorcraft, then the dynamicpressure will increase by a factor of 26 over the M 0.058(V 20 ms) case shown previously. To show the ability of boththe FishBAC concept and the current FSI analysis to cope with thisvery different operating condition, we will briefly consider theperformance of a new geometry configuration, which has beensuitably stiffened through an increase in the bending-spine thickness(tbs 4 mm) and the skin thickness (ts 2.5 mm). Additionally,the start of the morphing section has been moved back to 50% chord(xs 0.5) to allow for a larger rigid portion of the airfoil toaccommodate the wing-box structures typical of fixed-wingapplications. The remainder of the geometric and operating-pointparameters is the same as those detailed in Tables 1 and 2. Theconverged morphed shapes obtainable with this configuration areshown in Fig. 21, along with the internal structure drawn to scale.Note that larger spooling-pulley rotation angles are required than
in the V 20 ms case to achieve similar levels of trailing-edgedisplacement. This is due to increased elastic deformations in thetendons. If desired, stiffer tendons could also be used to reducerotation requirements and elastic losses. Indeed, all aspects of the
Fig. 16 Structural results: a) shear force, b) bending moment, c) slope, and d) displacement.
Fig. 17 Equilibrium deflected FishBAC shapes.
316 WOODS, DAYYANI, AND FRISWELL
geometry of this configuration could be improved through the use ofan optimization routine. The intention of presenting the current, fairlyarbitrary, geometry is to show the ability of the FSI code and theFishBAC concept to produce meaningful results at widely differentoperating conditions.Continuing to the aerodynamic performance of this configuration,
Fig. 22a shows the range of lift coefficients achievable by sweepingthrough spooling-pulley rotation angles, and Fig. 22b shows thecorresponding drag coefficients. As before, we see a wide rangeof achievable lift coefficients at this single angle of attack( 5 deg) due to the morphing camber airfoil. As expected, thereis an increase in drag with increasing lift, with a rapid increasebeyond cl 1.25.
Fig. 18 Convergence of aerodynamic coefficients: a) lift and b) drag.
Fig. 19 Evolution of aerodynamic coefficients with actuation: a) lift and b) drag.
Fig. 20 Actuation moments: a) moment vs spooling-pulley rotation, and b) lift vs moment.
Fig. 21 Equilibriummorphed shapes for theM 0.3 (V 102 ms)configuration.
WOODS, DAYYANI, AND FRISWELL 317
We can show the results of Fig. 22 in another way, which providesan additional insight into the performance of this FishBAC byplotting the lift-to-drag ratios corresponding to the range of operatinglift coefficients achievable, as is done in Fig. 23. This shows thetypical effect of adding camber to NACA airfoils where the highestlift-to-drag ratios are achieved at modestly high lift coefficients .Finally, we consider the actuation requirements of this higher
dynamic-pressure configuration. Figure 24 shows the actuationmoments both vs spooling-pulley rotation (Fig. 24a) and in terms ofthe resulting lift coefficients (Fig. 24b). It can be seen that thisFishBAC configuration responds in a similar manner to the V 20 ms case, with a linear increase in moment with rotation, but with
diminishing returns in the lift coefficient. Note, however, thesignificantly higher tendon moments required for a given liftcoefficient. For example, achieving a lift coefficient of cl 1.5withthe V 20 ms configuration requires 1.16 N m of torque,whereas the same lift coefficient for the V 102 ms con-figuration requires 9.4 N m of torque. This is the direct result of theincreased dynamic pressure and structural stiffness.This second test case for the FSI code developed here has shown
the predicted behavior of one potential geometry configuration foruse at higher flow speeds. The code was found to provide stablesolutions over a wide range of actuation inputs, and the resultingpredicted behavior of the FishBAC active-camber morphing airfoilshows good performance.
This work has presented the development of a highly coupled,partitioned FSI analysis of the FishBAC concept. An analyticalstructural model based on the EB beam theory is developed andshown to provide good levels of accuracy compared to theexperimental data and FEA. The XFOIL inviscid panel-method codeis used to solve for the aerodynamic pressure acting on the morphingstructure. The impact of the antagonistic tendon system used to drivethe morphing deflections is included through a linear elastic tendon-stiffness model coupled to the structural formulation. Because oflarge differences in stiffness between the aerodynamics, actuation,and structure, stable convergence of the solution is found to requirethe application of relaxation parameters at the fluidstructureinterface and the tendonstructure interface. While both fixed andAitken relaxation parameters are considered, a fixed relaxationparameter is used in the results presented here due to faster solution
Fig. 22 Evolution of aerodynamic coefficients with actuation (V 102 ms): a) lift and b) drag.
Fig. 23 Lift-to-drag ratio vs lift coefficient (V 102 ms).
Fig. 24 Actuation moments (V 102 ms): a) moment vs pulley rotation, and b) lift vs moment.
318 WOODS, DAYYANI, AND FRISWELL
convergence. The various results of the FSI analysis are then shownfor two representative geometries and aerodynamic operating points,including a low-speed wind-tunnel configuration and a medium-speed full-scale configuration representative of a fixed-wing UAV, awind turbine, or the retreating blade of a helicopter. In conclusion, theFSI analysis presented here is found to be a capable and robust low-fidelity tool for understanding and predicting the behavior of thenovel and promising FishBAC morphing airfoil concept.
The research leading to these results has received funding from theEuropean Research Council under the European Unions SeventhFramework Programme (FP/2007-2013)/European Research Coun-cil grant agreement number .
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