Effect of multiple engine placement on aeroelastic trim and stability of flying wing aircraft

Effect of Multiple Engine Placement on Aeroelastic

Trim and Stability of Flying Wing Aircraft

Pezhman Mardanpoura,1,∗, Phillip W. Richardsa,2, Omid Nabipoura,3,Dewey H. Hodgesa,4

aDaniel Guggenheim School of Aerospace EngineeringGeorgia Institute of Technology, Atlanta, Georgia 30332-0150

Abstract

Effects of multiple engine placement on flutter characteristics of a backswept

flying wing resembling the HORTEN IV are investigated using the code

NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Aircraft).

Four identical engines with defined mass, inertia, and angular momentum

are placed in different locations along the span with different offsets from

the elastic axis while fixing the location of the aircraft c.g. The aircraft ex-

periences body freedom flutter along with non-oscillatory instabilities that

∗Corresponding authorEmail addresses: [email protected] (Pezhman Mardanpour),

[email protected] (Phillip W. Richards), [email protected] (OmidNabipour), [email protected] (Dewey H. Hodges)

1PhD Candidate and Graduate Research Assistant, Student member, AIAA, APS andASME.

2Graduate Research Assistant, Student member, AIAA.3Visiting Scholar, Daniel Guggenheim School of Aerospace Engineering. Student mem-

ber, AIAA.4Professor, Fellow, AIAA and AHS; member, ASME.

Preprint submitted to Journal of Fluids and Structures August 28, 2013

originate from flight dynamics. Multiple engine placement increases flutter

speed particularly when the engines are placed in the outboard portion of

the wing (60% to 70% span), forward of the elastic axis, while the lift to drag

ratio is affected negligibly. The behavior of the sub- and supercritical eigen-

values is studied for two cases of engine placement. NATASHA captures a

hump body-freedom flutter with low frequency for the clean wing case, which

disappears as the engines are placed on the wings. In neither case is there

any apparent coalescence between the unstable modes. NATASHA captures

other non-oscillatory unstable roots with very small amplitude, apparently

originating with flight dynamics. For the clean-wing case, in the absence of

aerodynamic and gravitational forces, the regions of minimum kinetic energy

density for the first and third bending modes are located around 60% span.

For the second mode, this kinetic energy density has local minima around the

20% and 80% span. The regions of minimum kinetic energy of these modes

are in agreement with calculations that show a noticeable increase in flutter

speed if engines are placed forward of the elastic axis at these regions.

Keywords: Nonlinear Aeroelasticity, Follower Force, Effects of Engine

Placement, HALE Aircraft, Area of Minimum Kinetic Energy, NATASHA

2

1. Introduction

Flying wing aircraft are typically flexible lightweight aircraft with high aero-

dynamic performance. They may exhibit body-freedom flutter when the

short-period mode of the aircraft couples with the first symmetric elastic

bending and torsion mode; see Chipman et al. (1984), Gyorgy-Falvy (1960),

Love et al. (2005), and Myhra (1998). Due to the absence of a vertical tail,

a static flight dynamic instability, which involves the yawing rotation of the

aircraft in the horizontal plane, is usually captured in stability analyses and

suppressed by control systems of the aircraft; see Chipman et al. (1984), Love

et al. (2005), Moore (2010), Stenfelt and Ringertz (2009), and Myhra (1998).

High-aspect-ratio flying wings may undergo large deflections, which leads

to geometrically nonlinear behavior; see Patil and Hodges (2004). Previ-

ous studies by Patil and Hodges (2004), Patil et al. (2001), and Patil and

Hodges (2006) showed the inaccuracy of linear aeroelastic analysis and the

importance of nonlinear aeroelastic analysis. NATASHA (Nonlinear Aeroe-

lastic Trim And Stability of High Altitude Long Endurance Aircraft) is

the computer program used by Patil and Hodges (2006) and Chang et al.

(2008) for this study. It is based on the nonlinear composite beam theory of

Hodges (2003), which accommodates the modeling of high-aspect-ratio wings.

3

NATASHA uses the aerodynamic theory of Peters et al. (1995) and assesses

aeroelastic stability using the p method. Sotoudeh et al. (2010) presented

additional parametric studies using NATASHA primarily for the purposes

of verification and validation. However, neither the effects of sweep nor of

engine placement were included in these studies. Previous comparisons by

Sotoudeh et al. (2010) showed that results from NATASHA are in excellent

agreement with well-known beam stability solutions (Timoshenko and Gere,

1961; Simitses and Hodges, 2006), the flutter problem of Goland (1945),

experimental data presented by Dowell et al. (1977), and results from well-

established computer codes such as DYMORE (Bauchau and Kang, 1993;

Bauchau, 1998), and RCAS (Saberi et al., 2004). The behavior of sub- and

supercritical eigenvalues was verified by Mardanpour et al. (2012) using the

classical cantilever wing model of Goland (1945) and the continuum aero-

dynamics model of Balakrishnan (2012). In the same work, they studied

the suitability of modeling sweep with NATASHA using the same Goland

model. For the effect of sweep on divergence they compared results from

NATASHA with the approximate formula of Hodges and Pierce (2011), and

for flutter they compared results with work done by Lottati (1985). In both

cases results were in excellent agreement.

4

Effects of follower forces on dynamic instability of beams were studied

by Beck (1952); Bolotin (1959); Como (1966); Wohlhart (1971); Feldt and

Herrmann (1974). Despite engine thrust being a follower force, few studies

included this effect along with aeroelastic effects until the work of Hodges

et al. (2002), who presented a case in which the thrust vectors (from massless

engines) were placed on the outboard portion of the wings of an aircraft with

high aspect ratio wings, thus maximizing thrust effects. They concluded that

increasing engine thrust can either stabilize or destabilize, and flutter speed

and frequency were highly dependent on the ratio of bending stiffness and

torsional stiffness of the wing.

Fazelzadeh et al. (2009) studied the effect of a follower force and mass

arbitrarily placed along a long, straight, homogeneous wing. Their results

emphasize the effect of follower forces along with the external mass magni-

tude and location on the flutter characteristics. Elishakoff and Lottati (1988),

Karpouzian and Librescu (1993), and Fazelzadeh et al. (2009) studied the ef-

fect of sweep on flutter boundaries, but none of them used the geometrically

exact equations for beams. Mardanpour et al. (2012) studied the effect of

engine placement on nonlinear aeroelastic trim and stability of a flying wing

the geometry of which was similar to that of the Horten IV. They modeled

5

each engine as a rigid body with a mass, an inertia matrix, a thrust vector,

and a value of angular momentum. The result of their study showed that

the maximum flutter speed occurs when the engines are just outboard of

60% span; also, the minimum flutter speed occurs when the engine is placed

at wing tips. Both minima and maxima occurred when the engine c.g. was

located forward of the wing elastic axis. In their study, in the absence of

aerodynamics, gravitational forces, and engines, NATASHA found that the

minimum kinetic energy region for the first symmetric elastic free-free bend-

ing mode is near 60% span, which also coincides with the 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 trim and stability for two sets of engine placements

with different offsets from the elastic axis.

Nomenclature

a Deformed beam aerodynamic frame of reference

b Undeformed beam cross-sectional frame of reference

B Deformed beam cross-sectional frame of reference

6

bi 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 Chord

cmβ 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 0cT

e Offset of aerodynamic center from the origin of frame of reference along b2

f Column matrix of distributed applied force measures in Bi basis

F Column matrix of internal force measures in Bi basis

g Gravitational vector in Bi basis

H Column matrix of cross-sectional angular momentum measures in Bi basis

i Inertial frame of reference

ii Unit vectors for inertial frame of reference (i = 1, 2, 3)

I Cross-sectional inertia matrix

k Column matrix of undeformed beam initial curvature and twist measures in bi basis

K Column matrix of deformed beam curvature and twist measures in Bi basis

m Column matrix of distributed applied moment measures in Bi basis

M Column matrix of internal moment measures in Bi basis

7

P Column matrix of cross-sectional linear momentum measures in Bi basis

r Column matrix of position vector measures in bi basis

u Column matrix of displacement vector measures in bi basis

V Column matrix of velocity measures in Bi basis

x1 Axial coordinate of beam

β Trailing edge flap angle

γ Column matrix of 1-D generalized force strain measures

∆ Identity matrix

κ Column matrix of elastic twist and curvature measures (1-D generalized moment strain measures)

λ0 Induced flow velocity

µ Mass per unit length

ξ Column matrix of center of mass offset from the frame of reference origin in bi basis

ψ Column matrix of small incremental rotations

Ω 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

8

2. Theory

2.1. 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 form of nonlinear composite beam theory of Hodges

(2003) is based on first-order partial differential equations of motion for the

beam that are independent of displacement and rotation variables. They

contain variables that are expressed in the bases of the reference frames 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

9

velocity with nonlinearities of 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 by the structural constitutive equations

γ

κ

=

R S

ST T

FB

MB

(2)

and the inertial constitutive equations

PB

HB

=

µ∆ −µξ

µξ I

VB

ΩB

. (3)

Finally, strain- and velocity-displacement equations are used to derive the

fully intrinsic kinematical partial differential equations of Hodges (2003),

which are given as

V ′B + KBVB + (e1 + γ)ΩB = γ,

Ω′B + KBΩB = κ.

(4)

10

In this set of equations, FB and MB are column matrices of cross-sectional

stress and moment resultant measures in the B frame, respectively; VB and

ΩB are column matrices of cross-sectional frame velocity and angular velocity

measures in the B frame, respectively; PB and HB are column matrices

of cross-sectional linear and angular momentum measures in the B 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 ξ3cT with 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 the partial

derivative with respect to time; and ( )′ denotes the partial derivative with

respect to the axial coordinate, x1. More details about these equations can

be found in Hodges (2006).

This is a complete set of first-order, partial differential equations. To

solve this complete set 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, and because displacement

11

and rotation variables do not appear, singularities caused by finite rotations

are avoided.

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

processing operations by integrating

r′i = Cibe1

(ri + ui)′ = CiB(e1 + γ)

(5)

and

(Cbi)′ = −kCbi

(CBi)′ = −(k + κ)CBi

(6)

2.2. Finite state inflow model of Peters et al.

The two-dimensional finite state aerodynamic model of Peters et al. (1995)

is a state-space, thin-airfoil, inviscid, incompressible approximation of an

infinite-state representation of the aerodynamic loads, which accounts for

induced flow in the wake and apparent mass effects, using known airfoil pa-

rameters. It accommodates large motion of the airfoil as well as deflection of

a small trailing-edge flap. Although the two-dimensional version of this the-

ory does not account for three-dimensional effects associated with the wing

12

tip, published data (Peters et al., 1995; Sotoudeh et al., 2010; Mardanpour

et al., 2012) show this theory is an excellent choice for approximation of

aerodynamic loads acting on high-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)

Maero = 2ρb[(cm0 + cmβ

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

(9)

where

VT =√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 deflec-

tion.

The effect of unsteady wake (inflow) and apparent mass appear as λ0 and

13

acceleration terms in the force and moment equation. The inflow model of

Peters et al. (1995) is included to calculate λ0 as:

λ0 =1

2binflowTλ (13)

where λ is the column matrix of inflow states and satisfies

[Ainflow] λ+

(VTb

)λ =

(−Va3 +

b

2Ωa1

)cinflow (14)

and where [Ainflow], binflow, and cinflow are constant matrices derived by

Peters et al. (1995).

3. NATASHA

NATASHA is based on a geometrically exact formulation of the composite

beam theory of Hodges (2006) and the finite-state inflow aerodynamic model

of Peters et al. (1995). The governing equations for structural model are

geometrically exact, fully intrinsic and capable of analyzing the dynamical

behavior of a general, nonuniform, twisted, curved, anisotropic beam under-

going large deformation. The partial differential equations’ dependence on

x1 is approximated by a spatial central differencing presented by the work

14

of Patil and Hodges (2006). The resulting nonlinear ordinary differential

equations are linearized about a static equilibrium state. The equilibrium

state is governed by nonlinear algebraic equations, which NATASHA solves

in obtaining the steady-state trim solution using the Newton-Raphson pro-

cedure; see Patil and Hodges (2006). This system of nonlinear aeroelastic

equations, when linearized about the resulting trim state, leads to a stan-

dard eigenvalue problem which NATASHA uses to analyze the stability of

the structure. NATASHA is also capable of time marching the nonlinear

aeroelastic system of equations using a schedule of the flight controls, which

may be obtained from sequential trim solutions.

4. Case Study

The geometry of the flying wing studied in this paper resembles the HORTEN

IV as described by Gyorgy-Falvy (1960), Myhra (1998), and Mardanpour

et al. (2012); see Fig. 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, is modeled using six ele-

ments and a lumped mass whose center lies in the aircraft plane of symmetry.

Four engines with varying placement along the span and a set of flaps dis-

15

tributed on the wings comprise the main components of the aircraft flight

control system. As shown in Figs. 2 and 16, η1 and η2 are the dimensionless

distances along b1, along which the engines are located on the right wing;

r1 and r2 are radial offsets of the engines from the elastic axis, normalized

by the maximum radial offset from the elastic axis, rnominal = 0.3 meters; θ1

and θ2 are the polar angles, with θn = 0 (n = 1, 2) pointing upstream. The

properties of engines are presented in Table 1. 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 symme-

try was displaced such that the c.g. was held constant at (0, -1, -0.1) meters

with respect to the reference point.

The aerodynamic properties of the wing vary linearly from root to tip;

see Table 2. 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

16

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 1: Properties of the engines

Aerodynamic coefficient at 25% chord root tipcl0 1.07× 10−1 5.5× 10−3

clα 6.9476 rad−1 6.9981 rad−1

clδ 4.2891 rad−2 4.3288 rad−2

cd0 5.82× 10−3 6.74× 10−3

cm0 1.13× 10−2 −2.4× 10−4

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

Table 2: Sectional aerodynamic properties of the wing

Beam Section (VABS) for sections with different chord lengths; see Cesnik

and Hodges (1997) and Yu et al. (2002, 2012). The middle portion of the

aircraft is treated as a rigid body with constant aerodynamic and inertia

properties equal to those at the wing root. Detailed sectional properties of

the wing can be found in Table 3; these properties were tuned such that the

aircraft experiences body-freedom-flutter with the frequencies close to those

of the body-freedom flutter frequency obtained from HORTEN IV pilots as

described by Gyorgy-Falvy (1960), Myhra (1998), and Mardanpour et al.

(2012). See Table 4.

17

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 sectional moments of inertia:About the b1 axis 1.0132 kg-mAbout the b2 axis 0.0303 kg-mAbout the b3 axis 0.9829 kg-m

Table 3: Sectional properties of the wing

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

η1=0, η2=0.5 40.8 9.560 23.02 1.257

Table 4: Flutter characteristics of the base model

5. Effects of Engine Placement on Lift to Drag Ratio

Lift to drag ratio analysis is done by calculating the ratio of the equivalent

forces, namely weight and total thrust. Weight of the aircraft is assumed

to be constant, and at constant speed (50 m/s) while the aircraft c.g. was

held constant, NATASHA calculated the total thrust for different engine

placements along the span, including offsets from the elastic axis. In this

analysis, for example, the first engine on the right wing is fixed at η1 with a

particular offset from the elastic axis, and the second engine on the right wing

is further out along the span but with the same offset from the elastic axis.

18

It was observed that engine placement along the span does not significantly

affect L/D of the aircraft; see Figs. 3 – 10, The small variation of L/D can

be attributed to the small static deflections of the wings as the engines are

moved along the span – not due to considerations of the engine aerodynamics.

Figure 11 shows the contour of L/D for varying the offset of the engines

from the elastic axis while fixing their span wise 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

19

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


20



21


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

22



23

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

6. Body Freedom Flutter Characteristics

The behavior of the sub- and supercritical eigenvalues was studied for two

cases of engine placement, 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 a flutter frequency of 9.560 rad/s

while the first symmetric bending mode of the wings couples with the aircraft

short-period mode; see Fig. 13. In the supercritical regime, as the speed

increases, the modal damping peaks and then returns to the stable region (a

so-called hump mode) and another mode becomes unstable; see Fig. 12. The

24

second case experienced flutter 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

oscillatory mode presented in case (a) remains stable; see Fig. 14. Instability

occurs at a higher speed (88 m/s). There was no apparent coalescence be-

tween the unstable mode of the aircraft and other modes at the point where

instability occurred; see Fig. 15. In both cases, NATASHA captured 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

Table 4 and are used for normalization of other results.

25

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

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

26

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

27

7. Effect of Multiple Engine Placement on Body Freedom Flutter

Four identical engines with known mass, moments of inertia, and angular

momentum are symmetrically placed along the span (i.e., in the b1 direction),

and the massless rigid engine mounts are offset from the elastic axis in the

plane of the wing cross section (i.e., along b2 and b3), while the engine

orientations are maintained; see Fig. 16. The engine offsets from the elastic

axis are 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 with different offsets from the elastic axis

while one of the engines was kept fixed in a particular location and the

other one moves along the span. It is to be noted that the variability of

flutter speed with engine location is primarily the result of thrust, angular

momentum and inertial properties of the engines rather than changes in the

aerodynamic loads caused by deformation of the wing.

28

Figure 16: Schematic view of the flying wing

When engines are placed along the span with no offset from the elastic

axis, a higher flutter speed is obtained when the second engine is placed at

the outer portion of the wing and the first engine is at an area between 50%

to 70% span; see Fig. 17. This region continues to exhibit high flutter speeds

as engines are placed forward of the elastic axis (i.e. θ is in the first and fourth

quadrant). When the engines are placed behind the elastic axis (i.e. θ is in the

second and third quadrant) there is no significant peak in the flutter speed,

and maximum flutter speed occurs mostly when both engines are closer to

29

the root of the wing; see Figs. 18 – 25. It should be noted that normalized

flutter speeds greater than 3 are beyond the incompressibility assumption in

the aerodynamic model used in NATASHA. The results in this regime of flow

cannot be trusted, but they could be used as an indication of how the trend

of flutter speed might change.

For engine placement forward of the elastic axis, the unstable mode asso-

ciated with the area with noticeable increase in flutter speed, i.e. 50% to 70%

span, contains motion of a first 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 trim solution is symmetric, the unstable mode is antisymmetric – a first

bending-torsion mode; see Figs. 21, 22 and 23. This is caused by excitations

from an antisymmetric flight dynamic mode.

30

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

31



32



33



34


The maximum fluctuation of the flutter speed appears to be when the

second engine is at 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 show the contour of normalized flutter

speed and frequency when the first engine is at 50% span and the second

at 80% span. For engine placement forward of the elastic axis, normalized

flutter speed increases while there is little change in normalized flutter fre-

quency. The same behavior was observed in flutter speed as the first engine

is moved toward the outboard portion of the wing – closer to the second

engine. However, there is a rapid increase in flutter frequency accompanied

35

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

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

36



37



Another comparison is done for the case when the first engine is at 10%

span while the second engine is moved outboard (i.e. 50% to 70% span).

38

Contours of flutter speed and flutter frequency are presented in Figs. 32 –

37. When the second engine is placed at 50% span, as engines are moved

forward of the elastic axis the 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; see Figs. 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.


39



40



41


8. Area of Minimum Kinetic Energy

In the absence of aerodynamics, gravitational force, and engines, NATASHA

was used to calculate the kinetic energy of the modes of the aircraft; see

Figs. 38 – 40. This analysis was done in order to find the region where the

kinetic energy reaches its minimum for the first three lowest frequency free-

free elastic bending modes. Thus, the area of minimum kinetic energy for

the first and third bending modes is located around 60% span; see Figs. 38

and 40. For the second mode, the kinetic energy has local minima at 20%

and 80% span; see Fig. 39.

For engine placement forward of the elastic axis, the unstable mode con-

42

tains a combination of the 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 of minimum 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

43

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

wing

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

wing

44

9. Concluding Remarks

The aeroelastic trim and stability of a flying-wing aircraft with four en-

gines was analyzed. The aircraft model had a geometry similar to that of

the Horten IV with the addition of four identical engines of specified mass,

moments of inertia, and angular momentum. The four engines are symmet-

rically moved along the span with offsets from the elastic axis while fixing

the location of aircraft c.g. For the clean wing case, the aircraft experiences

body freedom flutter at 40.8 m/s with a flutter frequency of 9.560 rad/s. The

behavior of sub- and supercritical eigenvalues was studied for two cases of

engine placement, both with zero offset from the elastic axis but with differ-

ent locations along the span. The clean-wing case experiences a hump-mode

flutter, and as the engines are moved toward the outer portion of the wing the

unstable mode contains a combination of the first, second, and third bend-

ing along with torsion and the aircraft short period mode. In this case, the

hump-mode is not 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 lift to drag ratio. However, a noticeable increase in flutter

45

speed is observed when engines are placed forward of the elastic axis. For

these cases, as one of the engines is placed at the outboard portion of the

span, flutter speed increases. For engine placement behind the elastic axis,

flutter speed increases when both engines are close to the root.

In the absence of aerodynamics, gravitational force, and engines, the area

of minimum kinetic energy for the first and third bending modes is located

at approximately 60% span. For the second mode, the kinetic energy has

local minima around 20% and 80% span. The areas of minimum kinetic

energy for these modes are in agreement with flutter calculations, which

show a noticeable increase in flutter speed if engines are placed forward of

the elastic axis at these regions.

10. Acknowledgments

This work was supported in part by award ECCS-1101431 from the National

Science Foundation.

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Effect of multiple engine placement on aeroelastic trim and stability of flying wing aircraft

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