[American Institute of Aeronautics and Astronautics 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Boston, Massachusetts ()] 54th AIAA/ASME/ASCE/AHS/ASC

Effect of Multiple Engine Placement onAeroelastic Trim and Stability of Flying

Wing Aircraft

Pezhman Mardanpour∗, Phillip W. Richards †,

Omid Nabipour‡, and Dewey H. Hodges§,

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

Effects of multiple engine placement on flutter characteristics of a back-swept flying wing resembling the HORTEN IV are investigated using thecode NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Air-craft). Four identical engines with defined mass, inertia, and angular mo-mentum are placed in different locations along the span with different offsetsfrom the elastic axis while fixing the location of the aircraft c.g. The aircraftexperiences body freedom flutter along with non-oscillatory instabilitiesthat originate from flight dynamics. Multiple engine placement increasesflutter speed particularly when the engines are placed in the outboard por-tion of the wing (60% to 70% span), forward of the elastic axis, while the liftto drag ratio is affected negligibly. The behavior of the sub- and supercrit-ical eigenvalues is studied for two cases of engine placement. NATASHAcaptures a hump body-freedom flutter with low frequency for the cleanwing case, which disappears as the engines are placed on the wings. Inneither case is there any apparent coalescence between the unstable modes.NATASHA captures other non-oscillatory unstable roots with very smallamplitude, apparently originating with flight dynamics. For the clean-wingcase, in the absence of aerodynamic and gravitational forces, the regions ofminimum kinetic energy density for the first and third bending modes arelocated around 60% span. For the second mode, this kinetic energy densityhas local minima around the 20% and 80% span. The regions of minimumkinetic energy of these modes are in agreement with calculations that show

∗Graduate Research Assistant, Daniel Guggenheim School of Aerospace Engineering. Student member,AIAA and ASME. Email: [email protected]†Graduate Research Assistant, Daniel Guggenheim School of Aerospace Engineering. Student member,

AIAA. Email: [email protected]‡Visiting Scholar, Daniel Guggenheim School of Aerospace Engineering. Student member, AIAA. Email:

[email protected]§Professor, Daniel Guggenheim School of Aerospace Engineering. Fellow, AIAA and AHS; member,

ASME. Email: [email protected] c© 2013 by P. Mardanpour, P. W. Richards, O. Nabipour, and D. H. Hodges. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.

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54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

April 8-11, 2013, Boston, Massachusetts

AIAA 2013-1571

Copyright © 2013 by P. Mardanpour, P. W. Richards, O. Nabipour, and D. H. Hodges. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

a noticeable increase in flutter speed at these regions if engines are placedforward of the elastic axis.

Nomenclature

a Deformed beam aerodynamic frame of referenceb Undeformed beam cross-sectional frame of referenceB Deformed beam cross-sectional frame of referencebi Unit vectors in undeformed beam cross-sectional frame of reference (i = 1, 2, 3)Bi Unit vectors of deformed beam cross-sectional frame of reference (i = 1, 2, 3)c Chordcmβ Pitch moment coefficient w.r.t. flap deflection (β)clα Lift coefficient w.r.t. angle of attack (α)clβ Lift coefficient w.r.t. flap deflection (β)e1 Column matrix b1 0 0cTe Offset of aerodynamic center from the origin of frame of reference along b2

f Column matrix of distributed applied force measures in Bi basisF Column matrix of internal force measures in Bi basisg Gravitational vector in Bi basisH Column matrix of cross-sectional angular momentum measures in Bi basisi Inertial frame of referenceii Unit vectors for inertial frame of reference (i = 1, 2, 3)I Cross-sectional inertia matrixk Column matrix of undeformed beam initial curvature and twist measures in bi basisK Column matrix of deformed beam curvature and twist measures in Bi basism Column matrix of distributed applied moment measures in Bi basisM Column matrix of internal moment measures in Bi basisP Column matrix of cross-sectional linear momentum measures in Bi basisr Column matrix of position vector measures in bi basisu Column matrix of displacement vector measures in bi basisV Column matrix of velocity measures in Bi basisx1 Axial coordinate of beamβ Trailing edge flap angle∆ Identity matrixγ Column matrix of 1D generalized force strain measuresκ Column matrix of elastic twist and curvature measures (1D generalized moment strain measures)λ0 Induced flow velocityµ Mass per unit lengthξ Column matrix of center of mass offset from the frame of reference origin in bi basisψ Column matrix of small incremental rotationsΩ Column matrix of cross-sectional angular velocity measures in Bi basis( )′ Partial derivative of ( ) with respect to x1˙( ) Partial derivative of ( ) with respect to time

( ) Nodal variable

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American Institute of Aeronautics and Astronautics

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I. Introduction

Flying wing aircraft are typically flexible lightweight aircraft with high aerodynamic perfor-mance. They may exhibit body-freedom flutter when the short-period mode of the aircraftcouples with the first symmetric elastic bending and torsion mode.1,2, 3, 4 Due to the absenceof a vertical tail, a static flight dynamic instability, which involves the yawing rotation ofthe aircraft in the horizontal plane, is usually captured in stability analyses and suppressedby control systems of the aircraft.1,3, 5, 6, 4

High-aspect-ratio flying wings may undergo large deflections, which leads to geometricallynonlinear behavior.7 Previous studies by Hodges and Patil7,8, 9 showed the inaccuracy oflinear aeroelastic analysis and the importance of nonlinear aeroelastic analysis. NATASHA(Nonlinear Aeroelastic Trim And Stability of High Altitude Long Endurance Aircraft) isthe computer program9,10 used for this study. It is based on nonlinear composite beamtheory11 that accommodates the modeling of high-aspect-ratio wings. NATASHA uses theaerodynamic theory of Peters et al.,12 and assesses aeroelastic stability using the p method.Sotoudeh, Hodges and Chang13 presented additional parametric studies using NATASHAprimarily for the purposes of verification and validation. However, neither the effects of sweepnor of engine placement were included in these studies. Previous comparisons showed thatresults from NATASHA are in excellent agreement for the onset of instability.13 The behaviorof sub- and supercritical eigenvalues was verified by Mardanpour et al.14 using the classicalcantilever wing model of Goland15 and the continuum aerodynamics model of Balakrishnan.16

In the same work, they studied the suitability of modeling sweep with NATASHA using thesame Goland model. For the effect of sweep on divergence they compared results fromNATASHA with an approximate closed-form formula.17 For flutter they compared resultswith work done by Lottati,18 and in both cases results were in excellent agreement.

Effects of follower forces on dynamic instability of beams were studied by Refs. 19, 20,21,22,23. Despite engine thrust being a follower force, few studies included this effect alongwith aeroelastic effects until the work of Hodges et al.,24 who presented a case in which thethrust vectors (from massless engines) were placed on the outboard portion of the wings of anaircraft with high aspect ratio wings, thus maximizing thrust effects. They concluded thatincreasing engine thrust can either stabilize or destabilize, and flutter speed and frequencywere highly dependent on the ratio of bending stiffness and torsional stiffness of the wing.

Fazelzade et al.25 studied the effect of a follower force and mass arbitrarily placed along along, straight, homogeneous wing. Their results emphasize the effect of follower forces alongwith the external mass magnitude and location on the flutter characteristics. Lottati,26 Kar-pouzian and Librescu,27 and Mazidi et al.25 studied the effect of sweep on flutter boundaries,but none of them used the geometrically exact equations for beams. Mardanpour et al.14

studied the effect of engine placement on nonlinear aeroelastic trim and stability of a flyingwing the geometry of which was similar to that of the Horten IV. They modeled each engineas a rigid body with a mass, an inertia matrix, a thrust vector, and a value of angular mo-mentum. The result of their study showed that the maximum flutter speed occurs when theengines are just outboard of 60% span; also, the minimum flutter speed occurs the engine isplaced at wing tips. Both minima and maxima occurred when the engine c.g. was locatedforward of the wing elastic axis. In their study, in the absence of aerodynamics, gravitationalforces, and engines, NATASHA found that the minimum kinetic energy region for the firstsymmetric elastic free-free bending mode is near 60% span, which also coincides with the

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region where the maximum flutter speed was observed.In this paper, after briefly reviewing the theory behind NATASHA and explaining the

cases studied, behavior of the eigenvalues is examined along with an example of the trimand stability for two sets of engine placements with different offsets from the elastic offset.

II. Theory

A. Nonlinear composite beam theory

B 1

B 2

B 3

Deformed State

Undeformed State

r

R

R

s

ru

x1

b 1

b 2

b 3

R ˆ

r ˆ

Figure 1: Sketch of beam kinematics

The fully intrinsic nonlinear composite beam theory11 is based on first-order partialdifferential equations of motion for the beam that are independent of displacement androtation variables. They contain variables that are expressed in the bases of the referenceframes of the undeformed and deformed beams, b(x1) and B(x1, t), respectively; see Fig. 1.These equations are based on force, moment, angular velocity and velocity with nonlinearitiesof second order. The equations of motion are

F ′B + KBFB + fB = PB + ΩBPB

M ′B + KBMB + (e1 + γ)FB +mB = HB + ΩBHB + VBPB

(1)

where the generalized strains and velocities are related to stress resultants and moments bythe structural constitutive equations

γ

κ

=

[R S

ST T

]FBMB

(2)

and the inertial constitutive equationsPBHB

=

[µ∆ −µξµξ I

]VBΩB

. (3)

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Finally, strain- and velocity-displacement equations are used to derive the intrinsic kinemat-ical partial differential equations,11 which are given as

V ′B + KBVB + (e1 + γ)ΩB = γ

Ω′B + KBΩB = κ(4)

In this set of equations, FB and MB are column matrices of cross-sectional stress andmoment resultant measures in the B frame, respectively; VB and ΩB are column matrices ofcross-sectional frame velocity and angular velocity measures in the B frame, respectively; PBand HB are column matrices of cross-sectional linear and angular momentum measures in theB frame, respectively; R, S, and T are 3×3 partitions of the cross-sectional flexibility matrix;∆ is the 3×3 identity matrix; I is the 3×3 cross-sectional inertia matrix; ξ is b0 ξ2 ξ3cTwith ξ2 and ξ3 the position coordinates of the cross-sectional mass center with respect to the

reference line; µ is the mass per unit length; the tilde ( ) denotes the antisymmetric 3×3

matrix associated with the column matrix over which the tilde is placed; ˙( ) denotes thepartial derivative with respect to time; and ( )′ denotes the partial derivative with respectto the axial coordinate, x1. More details about these equations can be found in Ref. 28.

This is a complete set of first-order, partial differential equations. To solve this completeset of equations, one may eliminate γ and κ using Eq. (2) and PB and HB using Eq. (3).Then, 12 boundary conditions are needed, in terms of force (FB), moment (MB), velocity(VB) and angular velocity (ΩB). The maximum degree of nonlinearities is only two, andbecause displacement and rotation variables do not appear, singularities caused by finiterotations are avoided.

If needed, the position and the orientation can be calculated as post-processing operationsby integrating

r′i = Cibe1

(ri + ui)′ = CiB(e1 + γ)

(5)

and(Cbi)′ = −kCbi

(CBi)′ = −(k + κ)CBi(6)

B. Finite state inflow model of Peters et al.

The two-dimensional finite state aerodynamic model of Peters et al.12 is a state-space, thin-airfoil, inviscid, incompressible approximation of an infinite-state representation of the aero-dynamic loads, which accounts for induced flow in the wake and apparent mass effects, usingknown airfoil parameters. It accommodates large motion of the airfoil as well as deflectionof a small trailing-edge flap. Although the two-dimensional version of this theory does notaccount for three-dimensional effects associated with the wing tip, published data12,13,14

show this theory is an excellent choice for approximation of aerodynamic loads acting onhigh-aspect ratio wings.

The lift, drag and pitching moment at the quarter-chord are given by

Laero = ρb[(cl0 + clββ)VTVa2 − clαVa3b/2− clαVa2(Va3 + λ0 − Ωa1b/2)− cd0VTVa3

](7)

Daero = ρb[−(cl0 + clββ)VTVa3 + clα(Va3 + λ0)

2 − cd0VTVa2

](8)

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Maero = 2ρb[(cm0 + cmβ

β)VT − cmαVTVa3 − bclα/8Va2Ωa1 − b2clαΩa1/32 + bclαVa3/8]

(9)

whereVT =

√Va2 + Va3 . (10)

sinα =−Va3

VT(11)

αrot =Ωa1b/2

VT(12)

and Va2 , Va3 are the measure numbers of Va and β is the angle of flap deflection.The effect of unsteady wake (inflow) and apparent mass appear as λ0 and acceleration

terms in the force and moment equation. The inflow model of Peters et al.12 is included tocalculate λ0 as:

[Ainflow] λ+

(VTb

)λ =

(−Va3 +

b

2Ωa1

)cinflow (13)

λ0 =1

2binflowTλ (14)

where λ is the column matrix of inflow states, and [Ainflow], cinflow, binflow are constantmatrices derived in Ref. 12.

III. NATASHA

NATASHA is based on a geometrically exact formulation of composite beam theory28 and thefinite-state inflow aerodynamic model of Peters et al.12 The governing equations for structuralmodel are geometrically exact, fully intrinsic and capable of analyzing the dynamical behav-ior of a general, nonuniform, twisted, curved, anisotropic beam undergoing large deformation.The partial differential equations’ dependence on x1 is approximated by spatial central differ-encing.9 The resulting nonlinear ordinary differential equations are linearized about a staticequilibrium state. The equilibrium state is governed by nonlinear algebraic equations, whichNATASHA solves in obtaining the steady-state trim solution using the Newton-Raphson pro-cedure.9 This system of nonlinear aeroelastic equations, when linearized about the resultingtrim state, leads to a standard eigenvalue problem which NATASHA uses to analyze the sta-bility of the structure. NATASHA is also capable of time marching the nonlinear aeroelasticsystem of equations using a schedule of the flight controls, which may be obtained sequentialtrim solutions.

IV. Case Study

The geometry of the flying wing studied in this paper resembles the HORTEN IV;2,4, 14 seeFig. 2. This aircraft is modeled using 45 elements. Each wing is constructed with 19 elements;and the middle part of the aircraft, which accommodates a hypothetical pilot or cargo, ismodeled using six elements and a lumped mass whose center lies in the aircraft plane ofsymmetry. Four engines with varying placement along the span and a set of flaps distributedon the wings comprise the main components of the aircraft flight control system. As shownin Figs. 2 and 16, η1 and η2 are the dimensionless distances along b1 along which the engines

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are located on the right wing; r1 and r2 are radial offsets of the engines from the elasticaxis, normalized by the maximum radial offset from the elastic axis, rnominal = 0.3 meters.As the engines were placed further outboard along the span, the aircraft c.g. migrated aft.In order to counteract this effect, the concentrated mass in the aircraft plane of symmetrywas displaced such that the c.g. was held constant at (0, -1, -0.1) meters with respect to thereference point.

The aerodynamic properties of the wing vary linearly from root to tip; see Table 3 in theAppendix. The model properties vary from root to tip of the wings using these relations:

µ = µroot

(c

croot

)ξ = ξroot

(c

croot

)[R] = [R]root

(c

croot

)[S] = [S]root

(c

croot

)2

[I] = [I]root

(c

croot

)3

[T ] = [T ]root

(c

croot

)3

(15)

These relationships are derived empirically from the Variational Asymptotic Beam Section(VABS) for sections with different chord lengths.29,30,31 The middle portion of the aircraft istreated as a rigid body with constant aerodynamic and inertia properties equal to those at thewing root. Detailed sectional properties of the wing can be found in Table 2; these propertieswere tuned such that the aircraft experiences body-freedom-flutter with the frequencies closeto those of the body-freedom flutter frequency obtained from HORTEN IV pilots.2,4, 14 SeeTable 1.

V. Effects of Engine Placement on Lift to Drag Ratio

Lift to drag ratio analysis is done by calculating the ratio of the equivalent forces, namelyweight and total thrust. Weight of the aircraft is assumed to be constant, and at constantspeed (50 m/s) while the aircraft c.g. was held constant, NATASHA calculated the totalthrust for different engine placements along the span, including offsets from elastic axis. Inthis analysis, for example, the first engine on the right wing is fixed at η1 with a particularoffset from the elastic axis, and the second engine on the right wing is further out along thespan but with the same offset from the elastic axis. It was observed that engine placementalong the span does not affect L/D of the aircraft; see Figs. 3 – 10. Figure 11 shows thecontour of L/D for varying the offset of the engines from elastic axis while fixing their spanwise location; i.e., η1 = 0.1 and η2 = 0.3. It is shown that the change in L/D is not large.

Figure 2: Top view of the flying wing

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Figure 3: Lift to drag ratio for r1 = r2 = 0.3 and θ1 = θ2 = 0



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Figure 8: Lift to drag ratio for r1 = r2= 0.3 and θ1 = θ2= 225

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Figure 11: L/D contour for η1 = 0.1 and η2= 0.3

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VI. Body Freedom Flutter Characteristics

The behavior of the sub- and supercritical eigenvalues was studied for two cases of engineplacement, both with zero offset from the elastic axis: (a) η1 = 0 and η2 = 0.5, and (b)η1=0.6 and η2=0.9. For the first case, body-freedom flutter occurred at 40.8 m/s with aneigenvalue of 0.0006 ± 9.560i rad/s while the first symmetric bending mode of the wingscouples with the aircraft short-period mode; see Fig. 13. In the supercritical regime, as thespeed increases, the modal damping peaks and then returns to the stable region (a so-calledhump mode); another mode becomes unstable; see Fig. 12. The second case experiencedflutter at 88 m/s with frequency 47.12 rad/s. The unstable mode is a mixed motion of in-and out-of-plane bending coupled with torsion and the aircraft short-period mode.

As the engines are placed further outboard, i.e., case (b), the low-frequency oscillatorymode presented in case (a) remains stable; see Fig. 14. Instability occurs at a higher speed(88 m/s). There was no apparent coalescence between the unstable mode of the aircraft andother modes at the point where instability occurred; see Fig. 15. In both cases, NATASHAcaptured other non-oscillatory unstable roots of flight dynamic origin with very small am-plitude. The results at the onset of instability for case (a) are presented in Table1 and areused for normalization of other results.

Figure 12: Real part of the eigenvalues for η1=0 and η2=0.5

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Figure 13: Imaginary part of the eigenvalues for η1=0 and η2=0.5

Figure 14: Real part of the eigenvalues for η1=0.6 and η2=0.9

Figure 15: Imaginary part of the eigenvalues for η1=0.6 and η2=0.9

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Engine locations Speed (m/s) Eigenvalues (rad/s) Thrust (N) Flap (deg.)r1 = r2 = θ1 = θ2 = 0

η1=0, η2=0.5 40.8 0.0220.0006± 9.560i 23.02 1.257

Table 1: Flutter characteristics of the base model

VII. Effect of Multiple Engine Placement on Body FreedomFlutter

Four identical engines with known mass, moments of inertia, and angular momentum aresymmetrically placed along the span (i.e., in the b1 direction), and the engine mounts areoffset from the elastic axis in the plane of the wing cross section (i.e., along b2 and b3), whilethe engine orientations are maintained; see Fig. 16. The engine offsets from the elastic axisare presented in polar coordinates with (rn, θn) where n is the engine number. Figures 17– 25 show the variation in flutter speed for different engine placements along the span withdifferent offsets from the elastic axis while one of the engines was kept fixed in a particularlocation and the other one moves along the span.

Figure 16: Schematic view of the flying wing

When engines are placed along the span with no offset from the elastic axis, a higherflutter speed is obtained when the second engine is placed at the outer portion of the wingand the first engine is at an area between 50% to 70% span; see Fig. 17. This region continuesto exhibit high flutter speeds as engines are placed forward of the elastic axis (i.e. θ is inthe first and fourth quadrant). When the engines are placed behind the elastic axis (i.e. θis in the second and third quadrant) there is no significant peak in the flutter speed, and

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maximum flutter speed occurs mostly when both engines are closer to the root of the wing;see Figs. 18 – 25. It should be noted that normalized flutter speeds greater than 3 are beyondthe incompressibility assumption in the aerodynamic model used in NATASHA. The resultsin this regime of flow cannot be trusted, but they could be used as an indication on how thetrend of flutter speed might change.

For engine placement forward of the elastic axis, the unstable mode associated with thearea with noticeable increase in flutter speed, i.e. 50% to 70% span, contains motion of afirst bending-torsion coupled mode with second and third bending modes; see Figs. 17, 18,19, 20, and 25.

On the other hand, for engine placement behind the elastic axis, although the trimsolution is symmetric, the unstable mode is antisymmetric – a first bending-torsion mode;see Figs. 21, 22 and 23. This could be caused by excitations from an antisymmetric flightdynamic mode.

Figure 17: Normalized flutter speed for r1 = r2= 0 and θ1 = θ2= 0

Figure 18: Normalized flutter speed for r1 = r2= 0.3 and θ1 = θ2= 0

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The maximum fluctuation of the flutter speed appears to be when the second engine isat 80% span and the first engine is between 50% to 70%. To further investigate this area,contour plots of the flutter speed and frequency are presented in Figs. 26 – 31, which showthe contour of normalized flutter speed and frequency when the first engine is at 50% spanand the second at 80% span. For engine placement forward of the elastic axis, normalizedflutter speed increases while there is little change in normalized flutter frequency. The samebehavior was observed in flutter speed as first engine is moved toward the outboard portionof the wing – closer to the second engine. However, there is a rapid increase in flutterfrequency accompanied by a change in the unstable mode shape; see Figs. 28 – 31.

Figure 26: Normalized flutter speed contour for η1 = 0.5 and η2 = 0.8

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Figure 27: Normalized flutter frequency contour for η1 = 0.5 and η2 = 0.8



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Another comparison is done for the case when the first engine is at 10% span while thesecond engine is moved outboard (i.e. 50% to 70% span). Contours of flutter speed andflutter frequency are presented in Figs. 32 – 37. When the second engine is placed at 50%span, flutter speed increases and flutter frequency changes slightly; see Figs. 32 and 33.Placement of the second engine at 60% span increases the flutter speed to a higher range,and flutter frequency experiences a rapid change as the unstable mode shape changes; seeFigs. 34 and 35. When the second engine is placed farther outboard from the first (i.e. 70%),the flutter speed and frequency increase to higher values; see Figs. 36 and 37. In these cases,engine placement forward of the elastic axis increases the flutter speed.

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VIII. Area of Minimum Kinetic Energy

In the absence of aerodynamics, gravitational force, and engines, NATASHA was used tocalculate the area of minimum kinetic energy for the first, second and the third free-freebending modes of the aircraft; see Figs. 38 – 40. This analysis was done in order to findthe region where the kinetic energy reaches its minimum for the first three lowest frequencysymmetric elastic free-free bending modes. Thus, the area of minimum kinetic energy forthe first and third bending modes is located around 60% span; see Figs. 38 and 40. For thesecond mode, this area has a local minima at 20% and 80% span; see Fig. 39.

For engine placement forward of the elastic axis, the unstable mode contains a combina-tion of first, second, and third bending modes; and when the engines are placed around 60%to 80% span, there is a noticeable increase in flutter speed. This area is close to the area ofminimum kinetic energy of the first three bending modes; see Figs. 38 – 40.

Figure 38: Normalized kinetic energy of the symmetric first free-free mode of the flying wing

Figure 39: Normalized kinetic energy of the symmetric second free-free mode of the flyingwing

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Figure 40: Normalized kinetic energy of the symmetric third free-free mode of the flyingwing

IX. Concluding Remarks

The aeroelastic trim and stability of a flying-wing aircraft with four engines was analyzed.The aircraft model had a geometry similar to that of the Horten IV with the addition offour identical engines of specified mass, moments of inertia, and angular momentum. Thefour engines are symmetrically moved along the span with offsets from elastic axis whilefixing the location of aircraft c.g. For the clean wing case, the aircraft experiences bodyfreedom flutter at 40.8 m/s with an eigenvalue of 0.0006±9.560i rad/s. The behavior of sub-and supercritical eigenvalues was studied for two cases of engine placement, both with zerooffset from the elastic axis but with different locations along the span. The clean-wing caseexperiences a hump-mode flutter, and as the engines are moved toward the outer portionof the wing the unstable mode contains combination of the first, second, and third bendingalong with torsion and the aircraft short period mode. In this case, the hump-mode isnot unstable. In both cases, there is no apparent coalescence between the unstable modes.NATASHA also captures other non-oscillatory unstable roots from flight dynamics origin.

This study shows that engine placement does not have any significant effect on the liftto drag ratio. However, a noticeable increase in flutter speed is observed when engines areplaced forward of the elastic axis. For these cases, as one of the engines is placed at theoutboard portion of the span, flutter speed increases. For engine placement behind theelastic axis, flutter speed increases when both engines are close to the root.

In the absence of aerodynamics, gravitational force, and engines, the area of minimumkinetic energy for the first and third bending modes is located approximately 60% span.For the second mode, this area has local minima around 20% and 80% span. The areas ofminimum kinetic energy for these modes are in agreement with flutter calculations, whichshow noticeable increases in flutter speed when engines are placed in these regions forwardof the elastic axis.

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X. Appendix

Elastic axis (reference line) 25% chordAxial stiffness 1.162× 108 kg/s2

Torsional stiffness 1.883× 105 kg-m2

In-plane bending stiffness 1.349× 107 kg-m2

Out-of-plane bending stiffness 1.660× 105 kg-m2

Mass per unit length 9.193 kg/mInitial Curvature 1.70× 10−3/radMass offset −0.285 m

Wing inertias per unit span:About the b1 axis 1.0132 kg-mAbout the b2 axis 0.0303 kg-mAbout the b3 axis 0.9829 kg-m

Table 2: Sectional properties of the wing

root tipcl0 1.07× 10−1 rad−1 5.5× 10−3 rad−1

clα 6.9476 rad−1 6.9981 rad−1

clδ 4.2891 rad−2 4.3288 rad−2

cd0 5.82× 10−3 rad−1 6.74× 10−3 rad−1

cm0 1.13× 10−2 rad−1 −2.4× 10−4 rad−1

cmα −1.13× 10−2 rad−1 −8.3× 10−2 rad−1

cmδ−6.224× 10−1 rad−1 −7.276× 10−1 rad−1

Aerodynamic coefficient at 25% chord

Table 3: Sectional aerodynamic properties of the wing

Mass of each engine 10 kgAngular momentum of engine 5.829 kg-m2/s

Engine mass moment of inertia:About the b1 axis 0.3 kg-m2

About the b2 axis 0.3 kg-m2

About the b3 axis 0.3 kg-m2

Table 4: Properties of the engines

References

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[American Institute of Aeronautics and Astronautics 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Boston, Massachusetts ()] 54th AIAA/ASME/ASCE/AHS/ASC