Review Article On the Importance of Nonlinear ...downloads.hindawi.com/archive/2015/613962.pdf ·...

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


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]


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


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


Figure 10 Normalized flutter speed for 1199031= 1199032= 0 and 120595

1= 1205952=

0∘


0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




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


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


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


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


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)


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



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


[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


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



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





119861and 119867



moment (119872119861) velocity (119881


119861)



1199031015840

119894= 1198621198941198871198901

119903119894+ 1199061015840

119894= 119862119894119861(1198901+ 120574)

(119862119887119894)1015840

= minus 119862119887119894

(119862119861119894)1015840

= minus ( + 120581)119862119861119894

(5)





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)






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)


0as

[119860 inflow] + (119881119879

119887) 120582 = (minus

1198863+119887

2Ω1198861) 119888inflow

1205820=1

2119887inflow

119879

120582

(9)






120578

b3

b2

b1










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



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










120578

b3

b3b2

b2

b1


26

24

22

2

18

16

14

12

1

080 02 04 06 08 1

120578

uFu

F0



1

05

0

minus05

minus1


112

14 16

18 222

1

x3b

ave

x2bave


11511

105

1

095

09

085

08

08

075

075

075

1

05

0

minus05

minus1


x3b

ave

x2bave


b3

b3 b3

b3

b2b2

b2b2

b1b1

12057811205782

r1r2


35

3

25

2

15

0 02 04 06 08 1 12

1205781

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘


0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘

1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119861and 119867



moment (119872119861) velocity (119881


119861)



1199031015840

119894= 1198621198941198871198901

119903119894+ 1199061015840

119894= 119862119894119861(1198901+ 120574)

(119862119887119894)1015840

= minus 119862119887119894

(119862119861119894)1015840

= minus ( + 120581)119862119861119894

(5)





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)






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)


0as

[119860 inflow] + (119881119879

119887) 120582 = (minus

1198863+119887

2Ω1198861) 119888inflow

1205820=1

2119887inflow

119879

120582

(9)






120578

b3

b2

b1










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



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










120578

b3

b3b2

b2

b1


26

24

22

2

18

16

14

12

1

080 02 04 06 08 1

120578

uFu

F0



1

05

0

minus05

minus1


112

14 16

18 222

1

x3b

ave

x2bave


11511

105

1

095

09

085

08

08

075

075

075

1

05

0

minus05

minus1


x3b

ave

x2bave


b3

b3 b3

b3

b2b2

b2b2

b1b1

12057811205782

r1r2


35

3

25

2

15

0 02 04 06 08 1 12

1205781

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘


0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘

1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120578

b3

b2

b1










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



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










120578

b3

b3b2

b2

b1


26

24

22

2

18

16

14

12

1

080 02 04 06 08 1

120578

uFu

F0



1

05

0

minus05

minus1


112

14 16

18 222

1

x3b

ave

x2bave


11511

105

1

095

09

085

08

08

075

075

075

1

05

0

minus05

minus1


x3b

ave

x2bave


b3

b3 b3

b3

b2b2

b2b2

b1b1

12057811205782

r1r2


35

3

25

2

15

0 02 04 06 08 1 12

1205781

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘


0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘

1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120578

b3

b3b2

b2

b1


26

24

22

2

18

16

14

12

1

080 02 04 06 08 1

120578

uFu

F0



1

05

0

minus05

minus1


112

14 16

18 222

1

x3b

ave

x2bave


11511

105

1

095

09

085

08

08

075

075

075

1

05

0

minus05

minus1


x3b

ave

x2bave


b3

b3 b3

b3

b2b2

b2b2

b1b1

12057811205782

r1r2


35

3

25

2

15

0 02 04 06 08 1 12

1205781

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘


0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘

1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1

1205781

55

5

45

4

35

3

25

2

15

Nor

mal

ized

flut

ter s

peed




1= 1205952=

0∘

1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

08

06

04

02

00 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)




14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










14

12

1

08

06

04

0

02

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=1

)


1

09

08

07

06

05

0 02 04 06 08 1

120578

KE

(K

E) 120578

=0





1

09

08

07

06

05

04

03

02

010 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)


0

2

4

6

8

10

12

14

0 02 04 06 08 1

120578

KE

(120578)

KE

(120578=0

)






0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70

80

120578

0 02 04 06 08 1

KE

(120578)

KE

(120578=0

)



HorizonSunset

Eveningtwilight

Daylight





(ft)

96

96

180


120579

120579





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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











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)






8 Future Work



Nomenclature








119888 Chord119888119898120573







119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879


2


119894basis




119894basis







119894basis


119894basis


119894basis


119894basis





119894basis









119894basis


1




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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