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Review ArticleOn the Importance of Nonlinear Aeroelasticity andEnergy Efficiency in Design of Flying Wing Aircraft
Pezhman Mardanpour1 and Dewey H Hodges2
1University of California Los Angeles CA 90095 USA2Georgia Institute of Technology Atlanta GA 30332-0150 USA
Correspondence should be addressed to Dewey H Hodges dhodgesgatechedu
Received 17 August 2014 Accepted 4 December 2014
Academic Editor Jens N Sorensen
Copyright copy 2015 P Mardanpour and D H Hodges This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Energy efficiency plays important role in aeroelastic design of flying wing aircraft and may be attained by use of lightweightstructures as well as solar energy NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Aircraft) is a newly developedcomputer program which uses a nonlinear composite beam theory that eliminates the difficulties in aeroelastic simulations offlexible high-aspect-ratio wings which undergoes large deformation as well as the singularities due to finite rotations NATASHAhas shown that proper engine placement could significantly increase the aeroelastic flight envelope which typically leads to moreflexible and lighter aircraft The areas of minimum kinetic energy for the lower frequency modes are in accordance with the zoneswith maximum flutter speed and have the potential to save computational effort Another aspect of energy efficiency for HighAltitude Long Endurance (HALE) drones stems from needing to minimize energy consumption because of limitations on thesource of energy that is solar power NATASHA is capable of simulating the aeroelastic passivemorphingmaneuver (ie morphingwithout relying on actuators) and at as near zero energy cost as possible of the aircraft so as the solar panels installed on the wingare in maximum exposure to sun during different time of the day
1 Introduction
Over the last decade Hodges and coworkers [1ndash3] at GeorgiaTech have been extensively involved in aeroelastic simulationof very light and thus highly flexible aircraft for developmentof the next generation of unmanned aerial vehicles (UAVs)andor High-Altitude Long Endurance (HALE) aircraftincluding flying wings Such aircraft typically have high-aspect-ratio wings with high flexibility which leads to largedeformation Consequently linear aeroelastic analyses areincapable of predicting the stability characteristics of such air-craftThey successfully proved that only nonlinear aeroelasticanalysis provides correct information regarding the aeroelas-tic flight envelope of this class of aircraft [1 4 5]
Nonlinear aeroelastic trim and stability of HALE aircraftNATASHA is a computer program developed by the authorsof references [1 4 5] that accommodates modeling of largedeformation of high-aspect-ratio flying wings The theorybehind NATASHA is based on the geometrically exact
nonlinear composite beam theory of Hodges [6] along withthe finite-state induced flow model of Peters et al [7]
Previous comparisons by [2] showed that results fromNATASHA are in excellent agreement with well-known beamstability solutions [8 9] the flutter problem of [10] exper-imental data presented by [11] and results from well-estab-lished computer codes such as DYMORE [12 13] and RCAS[14] Mardanpour et al [3] considered the classical cantileverwing model of Goland and Luke [15] and verified NATASHAfor the behavior of the eigenvalues as well as the effect ofsweep on divergence and flutter characteristics and it wasshown that gravity and load factor play an important role inaeroelasticity of high-aspect-ratio wings [16]
A principal determinant of energy consumption in air-craft is drag which must be opposed by engine thrust for theaircraft to fly Flying wings may achieve significant dragreduction due to a smooth outer surface and the lack ofa vertical tail [3] Consequently the performance of suchaircraft may increase significantly relative to conventional
Hindawi Publishing CorporationAdvances in Aerospace EngineeringVolume 2015 Article ID 613962 11 pageshttpdxdoiorg1011552015613962
2 Advances in Aerospace Engineering
configurations of the same size The potential increase ofperformance for this class of aircraft has inspired aeroelas-ticians to design new generation of aircraft based on a flyingwing configuration [3] Typical aeroelastic instability of theseaircraft is body-freedom flutter when the short-period modeof the aircraft couples with the elastic bending-torsionmodes[3 17ndash24]
In another context a morphing solar-powered flyingwing can maximize the energy absorption of solar panelson the wing surfaces by changing its configuration such thatthe panels have highest exposure to the sun This change inthe geometry of the flying wing is highly effective in energyabsorption during times just before sunset and just aftersunrise and consequently the aircraft can sustain longer flight[25] Use of solar energy is a novelmethod that eliminates oneof the design constraints to a considerable extent by removingthe limitation on the source of energyThemorphing conceptcould be based on either wing morphing systems or airfoilmorphing systems or a combination of both [26] So far inthe literature several morphing concepts and systems havebeen developed based on altering various geometric param-eters of the wing (such as span chord camber sweep twistand even airfoil thickness distribution) to make the aircraftsuitable for different missions and flight conditions [19 2626 26ndash37]The folding wing configuration has been analyzedusing linear aeroelastic models [38ndash40] and nonlinear aeroe-lastic models [41 42] In all the mentioned works the weightof actuators and the actuation power that the morphingmechanisms require performing their task are the problem-atic parts of the design [26] in particular when it comes tomorphing of flying wing andor HALE aircraft
In this paper after a brief outline of the theory behindNATASHA energy efficiency in aeroelastic design and sim-ulation for flying wing configuration will be assessed as (a)a feature of the design (ie engine placement) which attainsinstability at higher speed with lighter aircraft structure (b) amethodology that helps to decrease computational effortrequired for determining favorable locations for engine place-ment with the potential of higher flutter speed and (c) ascheme to passively morph a solar-powered flying wing sothat exposure to the sun of solar panels distributed on thewings is maximized for higher absorption of solar energy atspecific times of the day
2 Theory
21 Nonlinear Composite Beam Theory The fully intrinsicnonlinear composite beam theory [6] is based on geometri-cally exact first-order partial differential equations of motionfor the beam that are independent of displacement androtation variables They contain variables that are expressedin the bases of the reference frames of the undeformed anddeformedbeams 119887(119909
1) and119861(119909
1 119905) respectively see Figure 1
These geometrically exact equations are written in termsof force moment velocity and angular velocity and theycontain no nonlinearities higher than second degree in theunknowns The equations of motion are
1198651015840
119861+ 119861119865119861+ 119891119861= 119861+ Ω119861119875119861
Undeformed state
Deformed state
x1
u
s
r
r
R
r
RR
b3b2
b1
B3B2
B1
Figure 1 Sketch of beam kinematics
1198721015840
119861+ 119861119872119861+ (1198901+ 120574) 119865
119861+ 119898119861
= 119861+ Ω119861119867119861+ 119861119875119861
(1)
where the generalized strains and velocities are related tostress resultants and moments by the structural constitutiveequations
120574
120581 = [
119877 119878
119878119879119879]
119865119861
119872119861
(2)
and the inertial constitutive equations
119875119861
119867119861
= [120583Δ minus120583120585
120583120585 119868]
119881119861
Ω119861
(3)
Finally strain- and velocity-displacement equations are usedto derive the intrinsic kinematical partial differential equa-tions [6] which are given as
1198811015840
119861+ 119861119881119861+ (1198901+ 120574)Ω
119861= 120574
Ω1015840
119861+ 119861Ω119861=
(4)
In this set of equations119865119861and119872
119861are columnmatrices of
cross-sectional stress and moment resultant measures in the119861 frame respectively 119881
119861and Ω
119861are column matrices of
cross-sectional frame velocity and angular velocity measuresin the 119861 frame respectively 119875
119861and119867
119861are column matrices
of cross-sectional linear and angular momentummeasures inthe 119861 frame respectively 119877 119878 and 119879 are 3 times 3 partitions ofthe cross-sectional flexibility matrix Δ is the 3 times 3 identitymatrix 119868 is the 3 times 3 cross-sectional inertia matrix 120585 islfloor0 12058521205853rfloor119879 with 120585
2and 1205853being position coordinates of the
cross-sectional mass center with respect to the reference lineand 120583 is the mass per unit length Note that the tilde notation( ) denotes the antisymmetric 3times3matrix associated with thecolumn matrix over which the tilde is placed ( ) denotes thepartial derivative with respect to time and ( )1015840 denotes the
Advances in Aerospace Engineering 3
partial derivative with respect to the axial coordinate 1199091
More details about these equations can be found in [43]This is a complete set of first-order partial differential
equations To solve this complete set of equations one mayeliminate 120574 and 120581 using (2) and 119875
119861and 119867
119861using (3) Then
12 boundary conditions are needed in terms of force (119865119861)
moment (119872119861) velocity (119881
119861) and angular velocity (Ω
119861)
The maximum degree of nonlinearities is only two andbecause displacement and rotation variables do not appearsingularities and other concerns related to finite rotation areavoided
If needed the position and the orientation can be calcu-lated as postprocessing operations by integrating
1199031015840
119894= 1198621198941198871198901
119903119894+ 1199061015840
119894= 119862119894119861(1198901+ 120574)
(119862119887119894)1015840
= minus 119862119887119894
(119862119861119894)1015840
= minus ( + 120581)119862119861119894
(5)
where 119862119894119887 is the transformation matrix from undeformedframe of reference 119887 to the inertial frame of reference 119894 and119862119894119861 is the transformation matrix from deformed frame of
reference 119861 to the inertial frame of reference 119894
22 Finite-State Induced Flow Model of Peters et al Thetwo-dimensional finite-state aerodynamic model of Peterset al [7] is a state-space thin-airfoil inviscid incompress-ible approximation of an infinite-state representation of theaerodynamic loads which accounts for induced flow in the
wake and apparent mass effects using known airfoil param-eters It accommodates large motion of the airfoil as well asdeflection of a small trailing edge flap (plusmn6∘) Although thetwo-dimensional version of this theory does not account forthree-dimensional effects associated with the wing tip pub-lished data [2 7] show that this theory is a suitable choice forrepresenting the aerodynamic loads acting on high-aspectratiowings Expressions for lift drag and pitchingmoment atthe quarter-chord are given by
119871aero = 1205881198871198812
119879(1198881198970+ 119888119897120572sin120572 + 119888
119897120573120573) + 120588119887119881
1198791198811198862119888119897120572120572rot cos120572
119863aero = 1205881198871198812
1198791198881198890+ 120588119887119881
1198791198811198862119888119897120572120572rot sin120572
119872aero = 212058811988721198812
119879(1198881198980+ 119888119898120572
sin120572 + 119888119898120573120573)
minus12058811988721198811198791198811198862119888119897120572120572rot
2
(6)where
119881119879= radic11988121198862+ 11988121198863
sin120572 =minus1198811198863
119881119879
120572rot =Ω11988611198872
119881119879
(7)
and 1198811198862 1198811198863are the measure numbers of 119881
119886and 120573 is the flap
deflection The effects of unsteady wake (induced flow) andapparent mass appear as 120582
0and acceleration terms in the
force and moment equations given as
119891119886= 120588119887
0
minus(1198881198970+ 119888119897120573120573)1198811198791198811198863+ 119888119897120572(1198811198863+ 1205820)2
minus 11988811988901198811198791198811198862
(1198881198970+ 119888119897120573120573)1198811198791198811198862minus1198881198971205721198863119887
2minus 1198881198971205721198811198862(1198811198863+ 1205820minusΩ1198861119887
2) minus 11988811988901198811198791198811198863
119898119886= 2120588119887
2
(1198881198980+ 119888119898120573120573)1198812
119879minus 1198881198981205721198811198791198811198863minus1198871198881198971205721198811198862Ω1198861
8minus1198872119888119897120572Ω1198861
32+1198871198881198971205721198863
80
0
(8)
The induced flow model of Peters et al [7] is included tocalculate 120582
0as
[119860 inflow] + (119881119879
119887) 120582 = (minus
1198863+119887
2Ω1198861) 119888inflow
1205820=1
2119887inflow
119879
120582
(9)
where 120582 is the column matrix of induced flow states and[119860 inflow] 119888inflow and 119887inflow are constant matrices derivedin [7]
3 Nonlinear Aeroelastic Trim and Stability ofHALE Aircraft (NATASHA)
NATASHA is a computer program that is based on a geomet-rically exact composite beam formulation [43] and a finite-state induced flow aerodynamic model [7] The governingequations for the structural model are geometrically exactfully intrinsic and capable of analyzing the dynamical behav-ior of a general nonuniform twisted curved anisotropicbeam undergoing large deformation The dependence of thepartial differential equations on 119909
1is approximated by spatial
4 Advances in Aerospace Engineering
120578
b3
b2
b1
Figure 2 Schematic view of the wing with engine
central differencing [5] A discretized version of the resultingnonlinear ordinary differential equations is linearized about astate of steady motion governed by nonlinear algebraic equa-tions which NATASHA solves to obtain the so-called trimsolution using Newton-Raphson or similar procedures [5]This system of nonlinear aeroelastic equations when lin-earized about the resulting trim state leads to a standardeigenvalue problem whichNATASHAuses to assess stability
4 Effect of Engine Placement onAeroelastic Flight Envelop
41 High-Aspect-Ratio Wing A lightweight small-classthrust turboshaft engine (JetCat SP5) [44] with known thrustmass moments of inertia and angular momentum whichoperates at the cruise condition is placed along the span of ahighly flexible high-aspect-ratio wing [16] The enginemounts are displaced with an offset from the elastic axisparallel to the plane of the wing cross section that is the localb2-b3plane while orientations of the engines aremaintained
see Figure 2In stability analysis with gravity (119892 = 98ms2) the
wing flutters at 224ms with a hump first bending-torsionmode and with a frequency of 12199 radsec which returnsto the stability region at higher speeds At 45ms the secondbending-torsion with a frequency of 27406 becomes unstable[16] NATASHArsquos result for flutter characteristics capturessignificant increase in flutter speedmore than three times thebaseline model (ie with engines at the root of the wing) forengines placed at 60 to 80 span and forward of the elasticaxis Also a significant decrease in flutter speed is observedfor engine placement at 30 to 50 span and aft of the elasticaxis Here forward means in the b
2direction and above the
elastic axis means in the b3direction see Figures 3 and 4
42 Two-Engine Flying Wing The geometry of the flyingwing resembles the HORTEN IV [18 20] and the propertieswere tuned [3] such that the aircraft experiences body-freedom flutter with the frequencies close to those of thebody-freedom flutter frequency obtained from interviewswith HORTEN IV pilots [20] Two identical engines weresymmetrically moved along the span that is in the b
1
0303
0606
0606
09
09 09
09
09
12
12
12
12
15
15
15
15
18
18
1821 21
2427333
0 02 04 06 08 1
1
05
0
minus05
minus1
x2b
x1l
Figure 3 Contour of normalized flutter speed for engine placementin chordwise direction that is b
2 considering gravity 119892 = 98ms2
02
02
04 04
06
06
08
08
08
1
1
1
1
12
12
12
12
1212
14
14
14
14
14
14
14
1616
16
0 02 04 06 08 1
1
05
0
minus05
minus1
x3b
x1l
Figure 4 Contour of normalized flutter speed for engine placementin the vertical direction that is b
3 considering gravity119892 = 98ms2
direction and displaced as described in Section 41 seeFigure 5
The effect of engine placement showed that themaximumflutter speed occurs when the engines are just outboard of60 span see Figure 6Theminimumflutter speed occurs forengine placement at the wing tips Both minima andmaximaoccurred when the cg of the engine was forward of the wingelastic axis [3] see Figures 7 and 8
43 Four-Engine FlyingWing The four-engine configuration(see Figure 9) is an alternative because of safety and relia-bility considerations associated with two-engine aircraftTheengine offsets from the elastic axis are presented in polarcoordinates with (119903
119899 120595119899) where 119899 is the engine number
Figures 10 and 11 show the variation in flutter speed for dif-ferent engine placements along the span with different offsetsfrom the elastic axis while one of the engines was kept fixed ina particular location and the other one moves along the span
Former study showed that engine placement does nothave a significant effect on the lift to drag ratio [24] Howevera noticeable increase in flutter speed is observed whenengines are placed forward of the elastic axis For these casesas one of the engines is placed at the outboard portion of thespan flutter speed increases A detailed study on this effect ispresented in the work of Mardanpour et al [24]
Advances in Aerospace Engineering 5
120578
b3
b3b2
b2
b1
Figure 5 Schematic view of the two-engine flying wing
26
24
22
2
18
16
14
12
1
080 02 04 06 08 1
120578
uFu
F0
Engine not offsetEngine forward = bave
Figure 6 Flutter speed for engine placement along the span for twovalues of chordwise offset
1
05
0
minus05
minus1
minus1 minus05 0 05 1
112
14 16
18 222
1
x3b
ave
x2bave
Figure 7 Contour of normalized flutter speed at 120578 = 065 two-engine flying wing
11511
105
1
095
09
085
08
08
075
075
075
1
05
0
minus05
minus1
minus1 minus05 0 05 1
x3b
ave
x2bave
Figure 8 Contour of normalized flutter speed at 120578 = 1 two-engineflying wing
b3
b3 b3
b3
b2b2
b2b2
b1b1
12057811205782
r1r2
Figure 9 Schematic view of the four-engine flying wing
35
3
25
2
15
0 02 04 06 08 1 12
1205781
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595
1= 1205952=
0∘
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
2 Advances in Aerospace Engineering
configurations of the same size The potential increase ofperformance for this class of aircraft has inspired aeroelas-ticians to design new generation of aircraft based on a flyingwing configuration [3] Typical aeroelastic instability of theseaircraft is body-freedom flutter when the short-period modeof the aircraft couples with the elastic bending-torsionmodes[3 17ndash24]
In another context a morphing solar-powered flyingwing can maximize the energy absorption of solar panelson the wing surfaces by changing its configuration such thatthe panels have highest exposure to the sun This change inthe geometry of the flying wing is highly effective in energyabsorption during times just before sunset and just aftersunrise and consequently the aircraft can sustain longer flight[25] Use of solar energy is a novelmethod that eliminates oneof the design constraints to a considerable extent by removingthe limitation on the source of energyThemorphing conceptcould be based on either wing morphing systems or airfoilmorphing systems or a combination of both [26] So far inthe literature several morphing concepts and systems havebeen developed based on altering various geometric param-eters of the wing (such as span chord camber sweep twistand even airfoil thickness distribution) to make the aircraftsuitable for different missions and flight conditions [19 2626 26ndash37]The folding wing configuration has been analyzedusing linear aeroelastic models [38ndash40] and nonlinear aeroe-lastic models [41 42] In all the mentioned works the weightof actuators and the actuation power that the morphingmechanisms require performing their task are the problem-atic parts of the design [26] in particular when it comes tomorphing of flying wing andor HALE aircraft
In this paper after a brief outline of the theory behindNATASHA energy efficiency in aeroelastic design and sim-ulation for flying wing configuration will be assessed as (a)a feature of the design (ie engine placement) which attainsinstability at higher speed with lighter aircraft structure (b) amethodology that helps to decrease computational effortrequired for determining favorable locations for engine place-ment with the potential of higher flutter speed and (c) ascheme to passively morph a solar-powered flying wing sothat exposure to the sun of solar panels distributed on thewings is maximized for higher absorption of solar energy atspecific times of the day
2 Theory
21 Nonlinear Composite Beam Theory The fully intrinsicnonlinear composite beam theory [6] is based on geometri-cally exact first-order partial differential equations of motionfor the beam that are independent of displacement androtation variables They contain variables that are expressedin the bases of the reference frames of the undeformed anddeformedbeams 119887(119909
1) and119861(119909
1 119905) respectively see Figure 1
These geometrically exact equations are written in termsof force moment velocity and angular velocity and theycontain no nonlinearities higher than second degree in theunknowns The equations of motion are
1198651015840
119861+ 119861119865119861+ 119891119861= 119861+ Ω119861119875119861
Undeformed state
Deformed state
x1
u
s
r
r
R
r
RR
b3b2
b1
B3B2
B1
Figure 1 Sketch of beam kinematics
1198721015840
119861+ 119861119872119861+ (1198901+ 120574) 119865
119861+ 119898119861
= 119861+ Ω119861119867119861+ 119861119875119861
(1)
where the generalized strains and velocities are related tostress resultants and moments by the structural constitutiveequations
120574
120581 = [
119877 119878
119878119879119879]
119865119861
119872119861
(2)
and the inertial constitutive equations
119875119861
119867119861
= [120583Δ minus120583120585
120583120585 119868]
119881119861
Ω119861
(3)
Finally strain- and velocity-displacement equations are usedto derive the intrinsic kinematical partial differential equa-tions [6] which are given as
1198811015840
119861+ 119861119881119861+ (1198901+ 120574)Ω
119861= 120574
Ω1015840
119861+ 119861Ω119861=
(4)
In this set of equations119865119861and119872
119861are columnmatrices of
cross-sectional stress and moment resultant measures in the119861 frame respectively 119881
119861and Ω
119861are column matrices of
cross-sectional frame velocity and angular velocity measuresin the 119861 frame respectively 119875
119861and119867
119861are column matrices
of cross-sectional linear and angular momentummeasures inthe 119861 frame respectively 119877 119878 and 119879 are 3 times 3 partitions ofthe cross-sectional flexibility matrix Δ is the 3 times 3 identitymatrix 119868 is the 3 times 3 cross-sectional inertia matrix 120585 islfloor0 12058521205853rfloor119879 with 120585
2and 1205853being position coordinates of the
cross-sectional mass center with respect to the reference lineand 120583 is the mass per unit length Note that the tilde notation( ) denotes the antisymmetric 3times3matrix associated with thecolumn matrix over which the tilde is placed ( ) denotes thepartial derivative with respect to time and ( )1015840 denotes the
Advances in Aerospace Engineering 3
partial derivative with respect to the axial coordinate 1199091
More details about these equations can be found in [43]This is a complete set of first-order partial differential
equations To solve this complete set of equations one mayeliminate 120574 and 120581 using (2) and 119875
119861and 119867
119861using (3) Then
12 boundary conditions are needed in terms of force (119865119861)
moment (119872119861) velocity (119881
119861) and angular velocity (Ω
119861)
The maximum degree of nonlinearities is only two andbecause displacement and rotation variables do not appearsingularities and other concerns related to finite rotation areavoided
If needed the position and the orientation can be calcu-lated as postprocessing operations by integrating
1199031015840
119894= 1198621198941198871198901
119903119894+ 1199061015840
119894= 119862119894119861(1198901+ 120574)
(119862119887119894)1015840
= minus 119862119887119894
(119862119861119894)1015840
= minus ( + 120581)119862119861119894
(5)
where 119862119894119887 is the transformation matrix from undeformedframe of reference 119887 to the inertial frame of reference 119894 and119862119894119861 is the transformation matrix from deformed frame of
reference 119861 to the inertial frame of reference 119894
22 Finite-State Induced Flow Model of Peters et al Thetwo-dimensional finite-state aerodynamic model of Peterset al [7] is a state-space thin-airfoil inviscid incompress-ible approximation of an infinite-state representation of theaerodynamic loads which accounts for induced flow in the
wake and apparent mass effects using known airfoil param-eters It accommodates large motion of the airfoil as well asdeflection of a small trailing edge flap (plusmn6∘) Although thetwo-dimensional version of this theory does not account forthree-dimensional effects associated with the wing tip pub-lished data [2 7] show that this theory is a suitable choice forrepresenting the aerodynamic loads acting on high-aspectratiowings Expressions for lift drag and pitchingmoment atthe quarter-chord are given by
119871aero = 1205881198871198812
119879(1198881198970+ 119888119897120572sin120572 + 119888
119897120573120573) + 120588119887119881
1198791198811198862119888119897120572120572rot cos120572
119863aero = 1205881198871198812
1198791198881198890+ 120588119887119881
1198791198811198862119888119897120572120572rot sin120572
119872aero = 212058811988721198812
119879(1198881198980+ 119888119898120572
sin120572 + 119888119898120573120573)
minus12058811988721198811198791198811198862119888119897120572120572rot
2
(6)where
119881119879= radic11988121198862+ 11988121198863
sin120572 =minus1198811198863
119881119879
120572rot =Ω11988611198872
119881119879
(7)
and 1198811198862 1198811198863are the measure numbers of 119881
119886and 120573 is the flap
deflection The effects of unsteady wake (induced flow) andapparent mass appear as 120582
0and acceleration terms in the
force and moment equations given as
119891119886= 120588119887
0
minus(1198881198970+ 119888119897120573120573)1198811198791198811198863+ 119888119897120572(1198811198863+ 1205820)2
minus 11988811988901198811198791198811198862
(1198881198970+ 119888119897120573120573)1198811198791198811198862minus1198881198971205721198863119887
2minus 1198881198971205721198811198862(1198811198863+ 1205820minusΩ1198861119887
2) minus 11988811988901198811198791198811198863
119898119886= 2120588119887
2
(1198881198980+ 119888119898120573120573)1198812
119879minus 1198881198981205721198811198791198811198863minus1198871198881198971205721198811198862Ω1198861
8minus1198872119888119897120572Ω1198861
32+1198871198881198971205721198863
80
0
(8)
The induced flow model of Peters et al [7] is included tocalculate 120582
0as
[119860 inflow] + (119881119879
119887) 120582 = (minus
1198863+119887
2Ω1198861) 119888inflow
1205820=1
2119887inflow
119879
120582
(9)
where 120582 is the column matrix of induced flow states and[119860 inflow] 119888inflow and 119887inflow are constant matrices derivedin [7]
3 Nonlinear Aeroelastic Trim and Stability ofHALE Aircraft (NATASHA)
NATASHA is a computer program that is based on a geomet-rically exact composite beam formulation [43] and a finite-state induced flow aerodynamic model [7] The governingequations for the structural model are geometrically exactfully intrinsic and capable of analyzing the dynamical behav-ior of a general nonuniform twisted curved anisotropicbeam undergoing large deformation The dependence of thepartial differential equations on 119909
1is approximated by spatial
4 Advances in Aerospace Engineering
120578
b3
b2
b1
Figure 2 Schematic view of the wing with engine
central differencing [5] A discretized version of the resultingnonlinear ordinary differential equations is linearized about astate of steady motion governed by nonlinear algebraic equa-tions which NATASHA solves to obtain the so-called trimsolution using Newton-Raphson or similar procedures [5]This system of nonlinear aeroelastic equations when lin-earized about the resulting trim state leads to a standardeigenvalue problem whichNATASHAuses to assess stability
4 Effect of Engine Placement onAeroelastic Flight Envelop
41 High-Aspect-Ratio Wing A lightweight small-classthrust turboshaft engine (JetCat SP5) [44] with known thrustmass moments of inertia and angular momentum whichoperates at the cruise condition is placed along the span of ahighly flexible high-aspect-ratio wing [16] The enginemounts are displaced with an offset from the elastic axisparallel to the plane of the wing cross section that is the localb2-b3plane while orientations of the engines aremaintained
see Figure 2In stability analysis with gravity (119892 = 98ms2) the
wing flutters at 224ms with a hump first bending-torsionmode and with a frequency of 12199 radsec which returnsto the stability region at higher speeds At 45ms the secondbending-torsion with a frequency of 27406 becomes unstable[16] NATASHArsquos result for flutter characteristics capturessignificant increase in flutter speedmore than three times thebaseline model (ie with engines at the root of the wing) forengines placed at 60 to 80 span and forward of the elasticaxis Also a significant decrease in flutter speed is observedfor engine placement at 30 to 50 span and aft of the elasticaxis Here forward means in the b
2direction and above the
elastic axis means in the b3direction see Figures 3 and 4
42 Two-Engine Flying Wing The geometry of the flyingwing resembles the HORTEN IV [18 20] and the propertieswere tuned [3] such that the aircraft experiences body-freedom flutter with the frequencies close to those of thebody-freedom flutter frequency obtained from interviewswith HORTEN IV pilots [20] Two identical engines weresymmetrically moved along the span that is in the b
1
0303
0606
0606
09
09 09
09
09
12
12
12
12
15
15
15
15
18
18
1821 21
2427333
0 02 04 06 08 1
1
05
0
minus05
minus1
x2b
x1l
Figure 3 Contour of normalized flutter speed for engine placementin chordwise direction that is b
2 considering gravity 119892 = 98ms2
02
02
04 04
06
06
08
08
08
1
1
1
1
12
12
12
12
1212
14
14
14
14
14
14
14
1616
16
0 02 04 06 08 1
1
05
0
minus05
minus1
x3b
x1l
Figure 4 Contour of normalized flutter speed for engine placementin the vertical direction that is b
3 considering gravity119892 = 98ms2
direction and displaced as described in Section 41 seeFigure 5
The effect of engine placement showed that themaximumflutter speed occurs when the engines are just outboard of60 span see Figure 6Theminimumflutter speed occurs forengine placement at the wing tips Both minima andmaximaoccurred when the cg of the engine was forward of the wingelastic axis [3] see Figures 7 and 8
43 Four-Engine FlyingWing The four-engine configuration(see Figure 9) is an alternative because of safety and relia-bility considerations associated with two-engine aircraftTheengine offsets from the elastic axis are presented in polarcoordinates with (119903
119899 120595119899) where 119899 is the engine number
Figures 10 and 11 show the variation in flutter speed for dif-ferent engine placements along the span with different offsetsfrom the elastic axis while one of the engines was kept fixed ina particular location and the other one moves along the span
Former study showed that engine placement does nothave a significant effect on the lift to drag ratio [24] Howevera noticeable increase in flutter speed is observed whenengines are placed forward of the elastic axis For these casesas one of the engines is placed at the outboard portion of thespan flutter speed increases A detailed study on this effect ispresented in the work of Mardanpour et al [24]
Advances in Aerospace Engineering 5
120578
b3
b3b2
b2
b1
Figure 5 Schematic view of the two-engine flying wing
26
24
22
2
18
16
14
12
1
080 02 04 06 08 1
120578
uFu
F0
Engine not offsetEngine forward = bave
Figure 6 Flutter speed for engine placement along the span for twovalues of chordwise offset
1
05
0
minus05
minus1
minus1 minus05 0 05 1
112
14 16
18 222
1
x3b
ave
x2bave
Figure 7 Contour of normalized flutter speed at 120578 = 065 two-engine flying wing
11511
105
1
095
09
085
08
08
075
075
075
1
05
0
minus05
minus1
minus1 minus05 0 05 1
x3b
ave
x2bave
Figure 8 Contour of normalized flutter speed at 120578 = 1 two-engineflying wing
b3
b3 b3
b3
b2b2
b2b2
b1b1
12057811205782
r1r2
Figure 9 Schematic view of the four-engine flying wing
35
3
25
2
15
0 02 04 06 08 1 12
1205781
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595
1= 1205952=
0∘
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Aerospace Engineering 3
partial derivative with respect to the axial coordinate 1199091
More details about these equations can be found in [43]This is a complete set of first-order partial differential
equations To solve this complete set of equations one mayeliminate 120574 and 120581 using (2) and 119875
119861and 119867
119861using (3) Then
12 boundary conditions are needed in terms of force (119865119861)
moment (119872119861) velocity (119881
119861) and angular velocity (Ω
119861)
The maximum degree of nonlinearities is only two andbecause displacement and rotation variables do not appearsingularities and other concerns related to finite rotation areavoided
If needed the position and the orientation can be calcu-lated as postprocessing operations by integrating
1199031015840
119894= 1198621198941198871198901
119903119894+ 1199061015840
119894= 119862119894119861(1198901+ 120574)
(119862119887119894)1015840
= minus 119862119887119894
(119862119861119894)1015840
= minus ( + 120581)119862119861119894
(5)
where 119862119894119887 is the transformation matrix from undeformedframe of reference 119887 to the inertial frame of reference 119894 and119862119894119861 is the transformation matrix from deformed frame of
reference 119861 to the inertial frame of reference 119894
22 Finite-State Induced Flow Model of Peters et al Thetwo-dimensional finite-state aerodynamic model of Peterset al [7] is a state-space thin-airfoil inviscid incompress-ible approximation of an infinite-state representation of theaerodynamic loads which accounts for induced flow in the
wake and apparent mass effects using known airfoil param-eters It accommodates large motion of the airfoil as well asdeflection of a small trailing edge flap (plusmn6∘) Although thetwo-dimensional version of this theory does not account forthree-dimensional effects associated with the wing tip pub-lished data [2 7] show that this theory is a suitable choice forrepresenting the aerodynamic loads acting on high-aspectratiowings Expressions for lift drag and pitchingmoment atthe quarter-chord are given by
119871aero = 1205881198871198812
119879(1198881198970+ 119888119897120572sin120572 + 119888
119897120573120573) + 120588119887119881
1198791198811198862119888119897120572120572rot cos120572
119863aero = 1205881198871198812
1198791198881198890+ 120588119887119881
1198791198811198862119888119897120572120572rot sin120572
119872aero = 212058811988721198812
119879(1198881198980+ 119888119898120572
sin120572 + 119888119898120573120573)
minus12058811988721198811198791198811198862119888119897120572120572rot
2
(6)where
119881119879= radic11988121198862+ 11988121198863
sin120572 =minus1198811198863
119881119879
120572rot =Ω11988611198872
119881119879
(7)
and 1198811198862 1198811198863are the measure numbers of 119881
119886and 120573 is the flap
deflection The effects of unsteady wake (induced flow) andapparent mass appear as 120582
0and acceleration terms in the
force and moment equations given as
119891119886= 120588119887
0
minus(1198881198970+ 119888119897120573120573)1198811198791198811198863+ 119888119897120572(1198811198863+ 1205820)2
minus 11988811988901198811198791198811198862
(1198881198970+ 119888119897120573120573)1198811198791198811198862minus1198881198971205721198863119887
2minus 1198881198971205721198811198862(1198811198863+ 1205820minusΩ1198861119887
2) minus 11988811988901198811198791198811198863
119898119886= 2120588119887
2
(1198881198980+ 119888119898120573120573)1198812
119879minus 1198881198981205721198811198791198811198863minus1198871198881198971205721198811198862Ω1198861
8minus1198872119888119897120572Ω1198861
32+1198871198881198971205721198863
80
0
(8)
The induced flow model of Peters et al [7] is included tocalculate 120582
0as
[119860 inflow] + (119881119879
119887) 120582 = (minus
1198863+119887
2Ω1198861) 119888inflow
1205820=1
2119887inflow
119879
120582
(9)
where 120582 is the column matrix of induced flow states and[119860 inflow] 119888inflow and 119887inflow are constant matrices derivedin [7]
3 Nonlinear Aeroelastic Trim and Stability ofHALE Aircraft (NATASHA)
NATASHA is a computer program that is based on a geomet-rically exact composite beam formulation [43] and a finite-state induced flow aerodynamic model [7] The governingequations for the structural model are geometrically exactfully intrinsic and capable of analyzing the dynamical behav-ior of a general nonuniform twisted curved anisotropicbeam undergoing large deformation The dependence of thepartial differential equations on 119909
1is approximated by spatial
4 Advances in Aerospace Engineering
120578
b3
b2
b1
Figure 2 Schematic view of the wing with engine
central differencing [5] A discretized version of the resultingnonlinear ordinary differential equations is linearized about astate of steady motion governed by nonlinear algebraic equa-tions which NATASHA solves to obtain the so-called trimsolution using Newton-Raphson or similar procedures [5]This system of nonlinear aeroelastic equations when lin-earized about the resulting trim state leads to a standardeigenvalue problem whichNATASHAuses to assess stability
4 Effect of Engine Placement onAeroelastic Flight Envelop
41 High-Aspect-Ratio Wing A lightweight small-classthrust turboshaft engine (JetCat SP5) [44] with known thrustmass moments of inertia and angular momentum whichoperates at the cruise condition is placed along the span of ahighly flexible high-aspect-ratio wing [16] The enginemounts are displaced with an offset from the elastic axisparallel to the plane of the wing cross section that is the localb2-b3plane while orientations of the engines aremaintained
see Figure 2In stability analysis with gravity (119892 = 98ms2) the
wing flutters at 224ms with a hump first bending-torsionmode and with a frequency of 12199 radsec which returnsto the stability region at higher speeds At 45ms the secondbending-torsion with a frequency of 27406 becomes unstable[16] NATASHArsquos result for flutter characteristics capturessignificant increase in flutter speedmore than three times thebaseline model (ie with engines at the root of the wing) forengines placed at 60 to 80 span and forward of the elasticaxis Also a significant decrease in flutter speed is observedfor engine placement at 30 to 50 span and aft of the elasticaxis Here forward means in the b
2direction and above the
elastic axis means in the b3direction see Figures 3 and 4
42 Two-Engine Flying Wing The geometry of the flyingwing resembles the HORTEN IV [18 20] and the propertieswere tuned [3] such that the aircraft experiences body-freedom flutter with the frequencies close to those of thebody-freedom flutter frequency obtained from interviewswith HORTEN IV pilots [20] Two identical engines weresymmetrically moved along the span that is in the b
1
0303
0606
0606
09
09 09
09
09
12
12
12
12
15
15
15
15
18
18
1821 21
2427333
0 02 04 06 08 1
1
05
0
minus05
minus1
x2b
x1l
Figure 3 Contour of normalized flutter speed for engine placementin chordwise direction that is b
2 considering gravity 119892 = 98ms2
02
02
04 04
06
06
08
08
08
1
1
1
1
12
12
12
12
1212
14
14
14
14
14
14
14
1616
16
0 02 04 06 08 1
1
05
0
minus05
minus1
x3b
x1l
Figure 4 Contour of normalized flutter speed for engine placementin the vertical direction that is b
3 considering gravity119892 = 98ms2
direction and displaced as described in Section 41 seeFigure 5
The effect of engine placement showed that themaximumflutter speed occurs when the engines are just outboard of60 span see Figure 6Theminimumflutter speed occurs forengine placement at the wing tips Both minima andmaximaoccurred when the cg of the engine was forward of the wingelastic axis [3] see Figures 7 and 8
43 Four-Engine FlyingWing The four-engine configuration(see Figure 9) is an alternative because of safety and relia-bility considerations associated with two-engine aircraftTheengine offsets from the elastic axis are presented in polarcoordinates with (119903
119899 120595119899) where 119899 is the engine number
Figures 10 and 11 show the variation in flutter speed for dif-ferent engine placements along the span with different offsetsfrom the elastic axis while one of the engines was kept fixed ina particular location and the other one moves along the span
Former study showed that engine placement does nothave a significant effect on the lift to drag ratio [24] Howevera noticeable increase in flutter speed is observed whenengines are placed forward of the elastic axis For these casesas one of the engines is placed at the outboard portion of thespan flutter speed increases A detailed study on this effect ispresented in the work of Mardanpour et al [24]
Advances in Aerospace Engineering 5
120578
b3
b3b2
b2
b1
Figure 5 Schematic view of the two-engine flying wing
26
24
22
2
18
16
14
12
1
080 02 04 06 08 1
120578
uFu
F0
Engine not offsetEngine forward = bave
Figure 6 Flutter speed for engine placement along the span for twovalues of chordwise offset
1
05
0
minus05
minus1
minus1 minus05 0 05 1
112
14 16
18 222
1
x3b
ave
x2bave
Figure 7 Contour of normalized flutter speed at 120578 = 065 two-engine flying wing
11511
105
1
095
09
085
08
08
075
075
075
1
05
0
minus05
minus1
minus1 minus05 0 05 1
x3b
ave
x2bave
Figure 8 Contour of normalized flutter speed at 120578 = 1 two-engineflying wing
b3
b3 b3
b3
b2b2
b2b2
b1b1
12057811205782
r1r2
Figure 9 Schematic view of the four-engine flying wing
35
3
25
2
15
0 02 04 06 08 1 12
1205781
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595
1= 1205952=
0∘
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
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DistributedSensor Networks
International Journal of
4 Advances in Aerospace Engineering
120578
b3
b2
b1
Figure 2 Schematic view of the wing with engine
central differencing [5] A discretized version of the resultingnonlinear ordinary differential equations is linearized about astate of steady motion governed by nonlinear algebraic equa-tions which NATASHA solves to obtain the so-called trimsolution using Newton-Raphson or similar procedures [5]This system of nonlinear aeroelastic equations when lin-earized about the resulting trim state leads to a standardeigenvalue problem whichNATASHAuses to assess stability
4 Effect of Engine Placement onAeroelastic Flight Envelop
41 High-Aspect-Ratio Wing A lightweight small-classthrust turboshaft engine (JetCat SP5) [44] with known thrustmass moments of inertia and angular momentum whichoperates at the cruise condition is placed along the span of ahighly flexible high-aspect-ratio wing [16] The enginemounts are displaced with an offset from the elastic axisparallel to the plane of the wing cross section that is the localb2-b3plane while orientations of the engines aremaintained
see Figure 2In stability analysis with gravity (119892 = 98ms2) the
wing flutters at 224ms with a hump first bending-torsionmode and with a frequency of 12199 radsec which returnsto the stability region at higher speeds At 45ms the secondbending-torsion with a frequency of 27406 becomes unstable[16] NATASHArsquos result for flutter characteristics capturessignificant increase in flutter speedmore than three times thebaseline model (ie with engines at the root of the wing) forengines placed at 60 to 80 span and forward of the elasticaxis Also a significant decrease in flutter speed is observedfor engine placement at 30 to 50 span and aft of the elasticaxis Here forward means in the b
2direction and above the
elastic axis means in the b3direction see Figures 3 and 4
42 Two-Engine Flying Wing The geometry of the flyingwing resembles the HORTEN IV [18 20] and the propertieswere tuned [3] such that the aircraft experiences body-freedom flutter with the frequencies close to those of thebody-freedom flutter frequency obtained from interviewswith HORTEN IV pilots [20] Two identical engines weresymmetrically moved along the span that is in the b
1
0303
0606
0606
09
09 09
09
09
12
12
12
12
15
15
15
15
18
18
1821 21
2427333
0 02 04 06 08 1
1
05
0
minus05
minus1
x2b
x1l
Figure 3 Contour of normalized flutter speed for engine placementin chordwise direction that is b
2 considering gravity 119892 = 98ms2
02
02
04 04
06
06
08
08
08
1
1
1
1
12
12
12
12
1212
14
14
14
14
14
14
14
1616
16
0 02 04 06 08 1
1
05
0
minus05
minus1
x3b
x1l
Figure 4 Contour of normalized flutter speed for engine placementin the vertical direction that is b
3 considering gravity119892 = 98ms2
direction and displaced as described in Section 41 seeFigure 5
The effect of engine placement showed that themaximumflutter speed occurs when the engines are just outboard of60 span see Figure 6Theminimumflutter speed occurs forengine placement at the wing tips Both minima andmaximaoccurred when the cg of the engine was forward of the wingelastic axis [3] see Figures 7 and 8
43 Four-Engine FlyingWing The four-engine configuration(see Figure 9) is an alternative because of safety and relia-bility considerations associated with two-engine aircraftTheengine offsets from the elastic axis are presented in polarcoordinates with (119903
119899 120595119899) where 119899 is the engine number
Figures 10 and 11 show the variation in flutter speed for dif-ferent engine placements along the span with different offsetsfrom the elastic axis while one of the engines was kept fixed ina particular location and the other one moves along the span
Former study showed that engine placement does nothave a significant effect on the lift to drag ratio [24] Howevera noticeable increase in flutter speed is observed whenengines are placed forward of the elastic axis For these casesas one of the engines is placed at the outboard portion of thespan flutter speed increases A detailed study on this effect ispresented in the work of Mardanpour et al [24]
Advances in Aerospace Engineering 5
120578
b3
b3b2
b2
b1
Figure 5 Schematic view of the two-engine flying wing
26
24
22
2
18
16
14
12
1
080 02 04 06 08 1
120578
uFu
F0
Engine not offsetEngine forward = bave
Figure 6 Flutter speed for engine placement along the span for twovalues of chordwise offset
1
05
0
minus05
minus1
minus1 minus05 0 05 1
112
14 16
18 222
1
x3b
ave
x2bave
Figure 7 Contour of normalized flutter speed at 120578 = 065 two-engine flying wing
11511
105
1
095
09
085
08
08
075
075
075
1
05
0
minus05
minus1
minus1 minus05 0 05 1
x3b
ave
x2bave
Figure 8 Contour of normalized flutter speed at 120578 = 1 two-engineflying wing
b3
b3 b3
b3
b2b2
b2b2
b1b1
12057811205782
r1r2
Figure 9 Schematic view of the four-engine flying wing
35
3
25
2
15
0 02 04 06 08 1 12
1205781
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595
1= 1205952=
0∘
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
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Navigation and Observation
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International Journal of
Advances in Aerospace Engineering 5
120578
b3
b3b2
b2
b1
Figure 5 Schematic view of the two-engine flying wing
26
24
22
2
18
16
14
12
1
080 02 04 06 08 1
120578
uFu
F0
Engine not offsetEngine forward = bave
Figure 6 Flutter speed for engine placement along the span for twovalues of chordwise offset
1
05
0
minus05
minus1
minus1 minus05 0 05 1
112
14 16
18 222
1
x3b
ave
x2bave
Figure 7 Contour of normalized flutter speed at 120578 = 065 two-engine flying wing
11511
105
1
095
09
085
08
08
075
075
075
1
05
0
minus05
minus1
minus1 minus05 0 05 1
x3b
ave
x2bave
Figure 8 Contour of normalized flutter speed at 120578 = 1 two-engineflying wing
b3
b3 b3
b3
b2b2
b2b2
b1b1
12057811205782
r1r2
Figure 9 Schematic view of the four-engine flying wing
35
3
25
2
15
0 02 04 06 08 1 12
1205781
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595
1= 1205952=
0∘
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 Advances in Aerospace Engineering
0 02 04 06 08 1
1205781
55
5
45
4
35
3
25
2
15
Nor
mal
ized
flut
ter s
peed
Eng no 2 at 9000 of spanEng no 2 at 8000 of spanEng no 2 at 7000 of spanEng no 2 at 6000 of span
Eng no 2 at 5000 of spanEng no 2 at 4000 of spanEng no 2 at 3000 of spanEng no 2 at 2000 of span
Figure 11 Normalized flutter speed for 1199031= 1199032= 03 and 120595
1= 1205952=
0∘
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 12 Normalized kinetic energy of the first bending mode ofthe wing
5 Area of Minimum Kinetic EnergyDensity of the Mode
Engine placement at certain location has the potential toincrease the flutter speed in two ways One is the locationwhere lower frequency flutter mode could be relegated to ahigher frequency mode and the other is the location wherethe fluid structure interaction is decreased Both criteriacould be met at the area of minimum kinetic energy of themode In other words engine placement at the area of min-imum kinetic energy of the modes has the potential to
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 13 Normalized kinetic energy of the second bending modeof the wing
1
08
06
04
02
00 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 14 Normalized kinetic energy of the first torsion mode ofthe wing
decrease fluid-structure interaction and enforce the structureto flutter at a higher mode [16] In case of the high-aspect-ratio wing presented in Section 41 the area of minimumkinetic energy density of the first two bending and torsionmodes of the wing in the absence of engines gravitationaland aerodynamic forces is presented in Figures 12ndash15 For thefirst bending and torsion modes this area is a minimum atthe root of the wing see Figures 12ndash14 For the secondbendingmode the area of minimum kinetic energy density ofthemode has a localminimumoutboard of the 85 span andfor second torsion mode this minimum moves to the regionbetween 70 and 90 span see Figures 13ndash15 For engineplacement forward of the elastic axis the unstable mode con-tains a combination of first and second bending along withsecond torsion and when the engines are placed around 50
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Aerospace Engineering 7
14
12
1
08
06
04
0
02
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=1
)
Figure 15 Normalized kinetic energy of the second torsion modeof the wing
1
09
08
07
06
05
0 02 04 06 08 1
120578
KE
(K
E) 120578
=0
Figure 16 Normalized kinetic energy of the first elastic free-freecoupled bending-torsion mode of the flying wing
to 70 span there is a noticeable increase in flutter speedThis is close to the area of minimum kinetic energy of thesecond bending and torsion modes see Figures 13ndash15
To further explore this possibility the same analysis isperformed for the two-engine and four-engine configura-tions Figure 16 depicts the kinetic energy distribution of thesymmetric free-free mode of the bare wing for the case studypresented in Section 42 One can see that the point of theminimum kinetic energy is just outboard of 60 span [3]
In case of the four-engine configuration for engine place-ment forward of the elastic axis the unstable mode containsa combination of first second and third bending modes[24] and when the engines are placed around 60 to 80span there is a noticeable increase in flutter speedThis area isclose to the area of minimum kinetic energy of the first threebending modes see Figures 17 18 and 19
1
09
08
07
06
05
04
03
02
010 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 17 Normalized kinetic energy of the symmetric first free-free elastic bending mode of the flying wing
0
2
4
6
8
10
12
14
0 02 04 06 08 1
120578
KE
(120578)
KE
(120578=0
)
Figure 18 Normalized kinetic energy of the symmetric second free-free elastic bending mode of the flying wing
6 Passive Morphing of Solar PoweredFlying Wing Aircraft
HALE aircraft can achieve sustained uninterrupted flighttime if they use solar power Wing morphing of solar-poweredHALE aircraft can significantly increase solar energyabsorbency [25] An example of the kind ofmorphing consid-ered in this chapter requires the wings to fold so as to orienta solar panel to be hitmore directly by the sunrsquos rays at specifictimes of the day [25] see Figure 20
In this study solar-powered HALE flying wing aircraft aremodeled with three beams with lockable hinge connectionsThere were four identical engines with prescribed massinertia and angularmomentumplaced at outer portion of thebeams with zero offset from the elastic axis see Figure 21
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Advances in Aerospace Engineering
0
10
20
30
40
50
60
70
80
120578
0 02 04 06 08 1
KE
(120578)
KE
(120578=0
)
Figure 19 Normalized kinetic energy of the symmetric third free-free elastic bending mode of the flying wing
Dusk DawnMorningtwilight
HorizonSunset
Eveningtwilight
Daylight
Figure 20 Schematic view of the solar-powered flying wing mor-phing and sun position
Three sets of flaps are distributed on the wings one set (set 1)is on the side wings the middle beam was divided into equalportions for the two other sets of flaps (sets 2 and 3) seeFigures 21 and 22
Such aircraft are shown to be capable of morphingpassively so that the solar panels are oriented toward thesun using only input from flight control surfaces and enginethrust The analysis underlying NATASHA was extended toinclude the ability to simulate morphing of the aircraft into aldquoZrdquo configuration see Figure 22 Because of the ldquolongendurancerdquo feature of HALE aircraft suchmorphing needs tobe done without relying on actuators and at as near zeroenergy cost as possible
For simulation of this complete maneuver a scheduleof ldquotrim conditionsrdquo is solved using the trim analysis inNATASHA The resulting control history (see Figure 23) isthen interpolated from these solutions and input intoNATASHA simulation The stability analysis of the trim con-ditions showed that this aircraft is aeroelastically stable andthe instabilities are flight dynamics modes with very smallnonoscillatory eigenvalues [25]
(ft)
96
96
180
Figure 21 Geometry of the solar-powered flying wing
120579
120579
Figure 22 Schematic front view of the morphed configuration ofthe solar-powered flying wing
Detailed systematic processes for trim and time-marching for a passively morphing flying wing are presentedin [25]
7 Concluding Remarks
Energy efficiency in aeroelastic design and simulation offlyingwingsmay be enhanced by appropriate choice of designfeatures for example engine placement which attains insta-bility at higher speed with lighter aircraft structure For thispurpose a very flexible high-aspect-ratio wing which dis-plays geometrically nonlinear behavior in subsonic flow isanalyzed using NATASHA The effect of engine placementalong the span with various offsets from the elastic axis isstudied using a small-class lightweight thruster (eg JetCatSP5) operating at the cruise condition A significant decreasein flutter speed is observed for engine placement at 30 to50 span aft of the elastic axis that is in the negative b
2
direction and above the elastic axis that is in the positive b3
directionThe same effect for a two-engine flyingwing config-uration with geometry similar to the Horten IV was studiedand because of safety and reliability considerations associ-ated with two-engine aircraft the analogy was extended toa four-engine configuration For the case of a two-engine
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Aerospace Engineering 9
Flap set 1Flap set 2
Flap set 3
15
1
05
0
minus05
minus1
minus150 5 10 15 20 25 30 35 40 45
Fold angle (deg)
Ang
le o
f flap
s (de
g)
Figure 23 Deflection of flaps versus angle of morph
configuration the maximum flutter speed occurs when theengines are just outboard of 60 span The minimum flutterspeed occurs for engine placement at the wing tips Bothminima and maxima occurred when the cg of the enginewas forward of the wing elastic axis In case of a four-engine configuration a noticeable increase in flutter speed isobserved when engines are placed forward of the elastic axisFor these cases as one of the engines is placed at the outboardportion of the span flutter speed increases For engineplacement aft of the elastic axis flutter speed increases whenboth engines are close to the root
In the absence of engines and aerodynamic and gravi-tational forces NATASHA found that (a) for a high-aspect-ratio wing the minimum kinetic energy region of the secondbendingmode has a local maximum around 45 to 55 spanand a local minimum at 80 to 90 span and the second tor-sion mode has these minima at 30 to 40 and 65 to 75span (b) for a two-engine flying wing it is very near the 60span for the first symmetric elastic free-free bending modeand (c) for a four-engine configuration theminimum kineticenergy regions for the first and third bending modes arelocated at approximately 60 span and for the second modethe kinetic energy has local minima around 20 and 80span The areas of minimum kinetic energy for these modesare in agreement with flutter calculations which show anoticeable increase in flutter speed if engines are placedforward of the elastic axis at these regions Knowledge ofthe minimum kinetic energy region thus has some potentialto help the designer and could possibly lead to significantsavings in computational effort
The analysis underlying NATASHA was extended toinclude the ability to simulatemorphing of the aircraft using anew set of trim and kinematical differential equations Anexample of the kind of morphing considered in this studyrequires the wings to fold so as to orient solar panels to behit more directly by the sunrsquos rays at specific times of the dayBecause of the ldquolong endurancerdquo feature of HALE aircraftsuchmorphing needs to be donewith as near zero energy costas possible that is without relying on actuators at the hingesbut instead using aerodynamic forces controlled by flap
deployments and engine thrusts The three-wing solar-powered HALE aircraft morphs passively into a Z-shapedconfiguration while local bending moments are zeroed out atthe hinge connection points but with the hinges locked withthe fold angle held at zero degrees while the aircraft morphsThe morphing motion is brought to a stop before the hingesare again locked
8 Future Work
The present aeroelastic simulation of passive wing morphingneglects the effects of hinge stiffness and damping but theiraddition is planned for a later work to enhance the realismof the model The addition of hinge damping will allowenergy dissipation to be taken into account and it is expectedthat incorporation of a spring-restrained hinge may facilitatepassive morphing and gust response of the aircraft during themorphing phases Other recommendations for future studiesinclude addition of the 3D Peters aerodynamic model toaccount for moderate aspect-ratio wings and axial flow Suchan aerodynamic model will extend the validity of analysisdone by NATASHA to rotary wings
In the present effort the effect of engine aerodynamics hasnot been considered Explicit formulation of the same wouldrequire the use of 3D computational fluid dynamic (CFD)model At this present moment to the best of the authorrsquosknowledge accurate simulation of the engine aerodynamicswill retain the qualitative aspect of the current analysis butmight lead to some quantitative improvements Anotheraspect worthy of future investigation is to account for the flex-ibility of the engine nacelle and mount It will be interestingto outline whether consideration of their flexibility and theresulting dynamics affects the aeroelastic behavior of theaircraftAlso the inertial properties andflexibilities of controlsurfaces are not studied including those in NATASHA thatfacilitates the simulations
Nomenclature
119886 Deformed beam aerodynamic frame ofreference
119887 Undeformed beam cross-sectional frameof reference
119861 Deformed beam cross-sectional frame ofreference
b119894 Unit vectors in undeformed beam
cross-sectional frame of reference(119894 = 1 2 3)
B119894 Unit vectors of deformed beam
cross-sectional frame of reference(119894 = 1 2 3)
119888 Chord119888119898120573
Derivative of pitch moment coefficientwith respect to flap deflection (120573)
119888119897120572 Derivative of lift coefficient with respect to
angle of attack (120572)119888119897120573 Derivative of lift coefficient with respect to
flap deflection (120573)
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Advances in Aerospace Engineering
1198901 Column matrix lfloor1 0 0rfloor
119879
119890 Offset of aerodynamic center from theorigin of frame of reference along b
2
119891 Column matrix of distributed appliedforce measures in B
119894basis
119865 Column matrix of internal force measuresin B119894basis
g Gravitational vector in B119894basis
119867 Column matrix of cross-sectional angularmomentum measures in B
119894basis
119894 Inertial frame of referencei119894 Unit vectors for inertial frame of reference
(119894 = 1 2 3)119868 Cross-sectional inertia matrix119896 Column matrix of undeformed beam
initial curvature and twist measures in b119894
basis119870 Column matrix of deformed beam
curvature and twist measures in B119894basis
119898 Column matrix of distributed appliedmoment measures in B
119894basis
119872 Column matrix of internal momentmeasures in B
119894basis
119875 Column matrix of cross-sectional linearmomentum measures in B
119894basis
119903 Column matrix of position vectormeasures in b
119894basis
119877 3 times 3 partition of the cross-sectionalflexibility matrix
119878 3 times 3 partition of the cross-sectionalflexibility matrix
119879 3 times 3 partition of the cross-sectionalflexibility matrix
119906 Column matrix of displacement vectormeasures in b
119894basis
119881 Column matrix of velocity measures in B119894
basis1199091 Axial coordinate of beam
120573 Trailing edge flap angleΔ Identity matrix120574 Column matrix of 1D generalized force
strain measures120581 Column matrix of elastic twist and
curvature measures (moment strain)1205820 Induced flow velocity
120583 Mass per unit length120585 Column matrix of center of mass offset
from reference line in b119894basis
Ω Column matrix of cross-sectional angularvelocity measures in B
119894basis
( )1015840 Partial derivative of ( ) with respect to 119909
1
( ) Partial derivative of ( ) with respect to time( ) Nodal variable( )119865 At flutter point
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Air Force ResearchLaboratory Contract FA8650-06-D-3624000708 throughBihrle Applied Research Inc and Award ECCS-1101431 fromthe National Science Foundation
References
[1] M J Patil and D H Hodges ldquoOn the importance of aerody-namic and structural geometrical nonlinearities in aeroelasticbehavior of high-aspect-ratio wingsrdquo Journal of Fluids andStructures vol 19 no 7 pp 905ndash915 2004
[2] Z Sotoudeh DHHodges andC S Chang ldquoValidation studiesfor aeroelastic Trim and Stability analysis of highly flexibleaircraftrdquo Journal of Aircraft vol 47 no 4 pp 1240ndash1247 2010
[3] P Mardanpour D H Hodges R Neuhart and N GraybealldquoEngine placement effect on nonlinear trim and stability offlying wing aircraftrdquo Journal of Aircraft vol 50 no 6 pp 1716ndash1725 2013
[4] M J Patil D H Hodges and C E S Cesnik ldquoNonlinear aeroe-lasticity and flight dynamics of high-altitude long-enduranceaircraftrdquo Journal of Aircraft vol 38 no 1 pp 88ndash94 2001
[5] M J Patil and D H Hodges ldquoFlight dynamics of highly flexibleflying wingsrdquo Journal of Aircraft vol 43 no 6 pp 1790ndash17992006
[6] D H Hodges ldquoGeometrically-exact intrinsic theory fordynamics of curved and twisted anisotropic beamsrdquo AIAAJournal vol 41 no 6 pp 1131ndash1137 2003
[7] D A Peters S Karunamoorthy and W-M Cao ldquoFinite stateinduced flow models part I two-dimensional thin airfoilrdquoJournal of Aircraft vol 32 no 2 pp 313ndash322 1995
[8] S P Timoshenko and J M Gere Theory of Elastic StabilityMcGraw-Hill New York NY USA 2nd edition 1961
[9] G J Simitses and D H Hodges Fundamentals of StructuralStability Elsevier Boston Mass USA 2006
[10] M Goland ldquoThe utter of a uniform cantilever wingrdquo Journal ofApplied Mechanics vol 12 no 4 pp A197ndashA208 1945
[11] E H Dowell J Traybar and D H Hodges ldquoAn experimental-theoretical correlation study of non-linear bending and torsiondeformations of a cantilever beamrdquo Journal of Sound andVibration vol 50 no 4 pp 533ndash544 1977
[12] O A Bauchau and N K Kang ldquoMultibody formulation forhelicopter structural dynamic analysisrdquo Journal of the AmericanHelicopter Society vol 38 no 2 pp 3ndash14 1993
[13] O A Bauchau ldquoComputational schemes for flexible nonlinearmulti-body systemsrdquoMultibody System Dynamics vol 2 no 2pp 169ndash225 1998
[14] H Saberi M Khoshlahjeh R A Ormiston and M JRutkowski ldquoRCAS overview and application to advanced rotor-craft problemsrdquo in Proceedings of the 4th Decennial SpecialistsrsquoConference onAeromechanics pp 741ndash781 AmericanHelicopterSociety San Francisco Calif USA January 2004
[15] M Goland and Y L Luke ldquoThe utter of a uniformwing with tipweightsrdquo Journal of Applied Mechanics vol 15 no 1 pp 13ndash201948
[16] P Mardanpour D H Hodges and R Rezvani ldquoNonlinearaeroelasticity of high-aspect-ratio wings excited by time-dependent thrustrdquo Nonlinear Dynamics vol 75 no 3 pp 475ndash500 2014
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Advances in Aerospace Engineering 11
[17] R R F Chipman M Rimer and B Muniz ldquoBody-freedomutter of a 12-scale forward swept-wing model an experimentaland analytical studyrdquo Tech Rep NAS1-17102 NASA LangleyResearch Center 1984
[18] D Gyorgy-Falvy ldquoPerformance analysis of the lsquoHorten IVrsquoFlying Wingrdquo in Proceedings of the 8th OSTIV CongressCologne Germany June 1960
[19] M H Love P S Zink P A Wieselmann and H YoungrenldquoBody freedom flutter of high aspect ratio flying wingsrdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference pp 1808ndash1830AIAA Austin Tex USA April 2005
[20] D Myhra The Horten Brothers and Their All-Wing AircraftSchiffer MilitaryAviation History Schiffer Publications AtglenPa USA 1998
[21] J R Banerjee ldquoFlutter characteristics of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 21 no 9 pp 733ndash736 1984
[22] J R Banerjee ldquoFlutter modes of high aspect ratio taillessaircraftrdquo Journal of Aircraft vol 25 no 5 pp 473ndash476 1988
[23] J SchweigerO Sensburg andH J Berns ldquoAeroelastic problemsand structural design of a tailless CFC-sailplanerdquo in Proceedingsof the 2nd International Forum on Aeroelasticity and StructuralDynamics pp 457ndash467 Aachen Germany 1985
[24] P Mardanpour P W Richards O Nabipour and D H HodgesldquoEffect of multiple engine placement on aeroelastic trim andstability of flying wing aircraftrdquo Journal of Fluids and Structuresvol 44 pp 67ndash86 2014
[25] P Mardanpour and D H Hodges ldquoPassive morphing of flyingwing aircraft Z-shaped configurationrdquo Journal of Fluids andStructures vol 44 pp 17ndash30 2014
[26] P Gamboa J Vale F J P Lau andA Suleman ldquoOptimization ofamorphing wing based on coupled aerodynamic and structuralconstraintsrdquo AIAA Journal vol 47 no 9 pp 2087ndash2104 2009
[27] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011
[28] D Cadogan T Smith F Uhelsky and M MacKusick ldquoMor-phing wing development for compact package unmannedaerial vehiclesrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Palm Springs Calif USA 2004 AIAA Paper2004-1807
[29] A Simpson N Coulombe J Jacob and S Smith ldquoMorphingwing development for compact package unmanned aerial vehi-clesrdquo in Proceedings of the 46th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials ConferenceAIAA Austin Tex USA 2005
[30] M T Stubbs Morphing wing development for compact packageunmanned aerial vehicles [MS thesis] Virginia PolytechnicInstitute and State University 2003
[31] D H Lee and T A Weisshaar ldquoAeroelastic on a foldingwing configurationrdquo in Proceedings of the 46 th AIAAASMEASCEAHSASC Structures Structural Dynamics amp MaterialsConference pp 2452ndash2464 Austin Tex USA April 2005
[32] M P Snyder B Sanders F E Eastep andG J Frank ldquoVibrationand flutter characteristics of a folding wingrdquo in Proceedingsof the 46th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference pp 2421ndash2432 AIAAAustin Tex USA April 2005
[33] D R Bye and P D McClure ldquoDesign of a morphing vehiclerdquo inProceedings of the 48thAIAAASMEASCEAHSASC Structures
Structural Dynamics and Materials Conference pp 321ndash336AIAA Honolulu Hawaii USA April 2007
[34] T G Ivanco R C Scott M H Love S Zink and T AWeisshaar ldquoValidation of the lockheed Martin morphing con-cept with wind tunnel testingrdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics andMaterials Conference pp 6464ndash6480 AIAAHonolulu HawaiiUSA April 2007
[35] J S Flanagan R C Strutzenberg R BMyers and J E RodrianldquoDevelopment and flight testing of a morphing aircraftthe NextGen MFX-1rdquo in Proceedings of the 48th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA Paper 2007-1707 pp 73ndash75 AIAA AustinTex USA April 2007
[36] G R Andersen D L Cowan and D J Piatak ldquoAeroelasticmodeling analysis and testing of amorphingwing structurerdquo inProceedings of the 46thAIAAASMEASCEAHSASC StructuresStructural Dynamics and Materials Conference AIAA AustinTex USA April 2007
[37] N Gandhi A Jha J Monaco T M Seigler D Ward and D JInman ldquoIntelligent control of a morphing aircraftrdquo in Proceed-ings of the 46th AIAAASMEASCEAHSASC Structures Struc-tural Dynamics and Materials Conference AIAA Austin TexUSA 2007
[38] RWlezien GHorner AMcGowan et al ldquoThe aircraftmorph-ing programrdquo in Proceedings of the 44th AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference AIAA Norfolk Va USA 1998
[39] J R Wilson ldquoMorphing UAVs change the shape of warfarerdquoAerospace America vol 42 no 2 pp 28ndash29 2004
[40] J Dunn L Horta T Ivanco et al ldquoNasa contributions todarpa mas programrdquo in Proceedings of the Aerospace Flutter andDynamics Council Meeting NASA Langley Research CenterHampton Va USA May 2004
[41] P J Attar D Tang and E H Dowell ldquoNonlinear aeroelasticstudy for folding wing structuresrdquo AIAA Journal vol 48 no 10pp 2187ndash2195 2010
[42] I Wang S C Gibbs and E H Dowell ldquoAeroelastic model ofmultisegmented folding wings theory and experimentrdquo Journalof Aircraft vol 49 no 3 pp 911ndash921 2012
[43] D H Hodges Nonlinear Composite Beam Theory AIAAReston Va USA 2006
[44] ldquoJetcat spt5 turboshaft enginerdquo Tech Rep 2012 httpwwwjetcatusacomspt5html
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of