[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 Inertial and Constitutive Propertieson Body-freedom Flutter of a Flying Wing

Phillip W. Richards∗, Pezhman Mardanpour†,

Robert A. Herd‡, and Dewey H. Hodges§

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

The class of high altitude long endurance (HALE) aircraft often havevery slender wings with minimal structure. This leads to large trim de-flections and coupling between the vehicle flight dynamics and wing vi-brations. When this coupled behavior becomes unstable, it is referred toas body-freedom flutter (BFF). BFF behavior is dependent on the iner-tial and constitutive properties of the wing as well as the fuselage. Thisrelationship is explored for a typical flying wing aircraft representative ofthe Horten IV flying wing using an efficient yet rigorous analysis that re-lies on geometrically-exact, fully intrinsic beam equations and a finite-stateinduced flow model, implemented in the computer code NATASHA (Non-linear Aeroelastic Trim and Stability of HALE Aircraft). The wing inertialand stiffness properties were calculated using a realistic representative sec-tion using the powerful section analysis tool VABS. Trade studies on theBFF behavior were performed by varying the fuselage properties and theinternal wing structure and examining the effects on the flutter speed, flut-ter frequency, and flutter mode shape.

I. Introduction

There is a current interest in high altitude long endurance (HALE) aircraft, especially forapplications involving surveillance, communication, or atmospheric measurement. This classof aircraft nearly always utilizes very high aspect-ratio wings, as induced drag issues become

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

AIAA. Email: [email protected]‡Undergraduate Research Assistant, Daniel Guggenheim School of Aerospace Engineering. Student mem-

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

ASME. Email: [email protected] c© 2013 by P. W. Richards, P. Mardanpour, R. A. Herd, and D. H. Hodges. Published by the

American Institute of Aeronautics and Astronautics, Inc., with permission.

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American Institute of Aeronautics and Astronautics

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

April 8-11, 2013, Boston, Massachusetts

AIAA 2013-1840

Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

critical at very high altitudes. The wing structure for HALE aircraft is minimal to reducethe wing weight as much as possible, and as a result, these aircraft can experience very largewing deformations in trimmed flight. The large wing deformations exhibited by these air-craft lead to geometric nonlinearities that largely affect the aeroelastic analysis process.1,2, 3

Conventional aeroelastic analyses often decouple the wing vibrations from the vehicle flightdynamics, but aircraft with these high aspect-ratio wings and minimal structure have beenknown to experience instabilities characterized by interaction between the vehicle flight dy-namics and the structural vibrations. One explanation for this interaction is that the extremelength and low stiffness of the wings result in natural vibration frequencies on the order of theflight dynamics. In any case, aeroelastic analysis of these flight vehicles results in aeroelasticmode shapes that have strong components of wing vibration and vehicle body motion.

It has been shown that the study of mode shapes is fundamental to the study of flutterin fixed wing aircraft, and that the coalescence of two or more mode shapes frequently leadsto the flutter instability.4,5 When the vehicle rigid body degrees of freedom are involved, theflutter instability is often referred to as body freedom flutter (BFF). Some have suggestedthat the important flutter modes can be determined by looking for coalescence or near-coalescence of two neighboring modes.6 Several past studies have examined the effect ofvarious structural and physical parameters on the flutter characteristics of wings, includingmass, mass moment of inertia, elastic and torsional modulus, etc.7 However, the generalstructural properties are not the only important factor in examining the flutter instability– in addition, the flutter behavior of the wing is also affected by the interior structure andmodeling of the wing, including how it is attached to the fuselage.8,9

Although the dependence of this body-freedom flutter on some of the wing constitutiveproperties is recognized by the literature (especially bending and torsional stiffness), it isnot clear if other constitutive properties have the potential to affect BFF. Additionally, noattempt has been found to translate the increases in stiffness or other properties to increasesin the actual wing structure. The present paper examines the effects of actual increases ofwing structure in terms of the BFF behavior, and also examines the effect of varying fuselageor payload properties as well.

II. Theory

A very efficient intrinsic theory for analyzing high-aspect ratio aircraft is available and quitesuitable for the aeroelastic analysis of HALE aircraft.10 This theory retains all geometricnonlinearities, and its use of intrinsic variables completely eliminates the potential for sin-gularities. These fully intrinsic equations contain variables expressed in bases attached tothe undeformed reference frame b(x, t) and the deformed frame B(x, t), and can be writtenin compact matrix form:11

F ′B + KBFB + fb = PB + ΩBPB

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

(1)

The generalized strains and velocities can then be related to stress resultants and momentaby structural and inertial constitutive equations:

γ

κ

=

[R S

ST T

]FB

MB

PB

HB

=

[µ∆ −µξµξ I

]VBΩB

(2)

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The kinematics can then be described by partial differential equations:

V ′B + KBVB + (e1 + γ)Ω = γ

Ω′B + KBΩB = κ(3)

Equations 2 – 3 therefore present a complete set of first-order, partial differential equationssuitable for the analysis of high-aspect ratio aircraft. These equations have been implementedby the computer code NATASHA12 and this code has been validated by several furtherstudies.13,14

A. Cross-sectional Analysis

The high-aspect-ratio wing is beam-like. Beams have one dimension much larger than theother two (cross-sectional) dimensions. Other examples in the aircraft industry include he-licopter and wind-turbine rotor blades. Because of the complexity of the interior region ofsuch beam-like structures, their analysis and design may be thought to be best carried outusing three-dimensional (3D) finite-element analyses (FEA) given that such analyses pro-vide for high-fidelity modeling of complex geometries and accurate, reliable results. However,there are some obvious drawbacks: invariably all FEA tools are computationally expensivewhen compared to beam modeling tools, often by two to three orders of magnitude. An-other aspect often overlooked is how labor-intensive even the modeling process can become,especially for complex layups and geometries. An obvious and popular alternative has beenconventional beam modeling techniques. Although they are computationally less expensive,the results are seldom satisfactory, especially for composite structures, which are frequentlythe cases of interest. An ideal methodology would combine the relatively inexpensive na-ture of beam modeling tools with the ability to achieve high-fidelity in modeling proceduresa la finite-element analyses, resulting in an efficient, reliable analysis tool with no ad hockinematic assumptions typically associated with standard beam modeling tools.

Such an asymptotically exact methodology, called the Variational Asymptotic Beam Sec-tion Analysis (VABS), has been developed over the last two decades with the objective tocreate the best possible set of elastic constants for an equivalent beam analysis from a de-tailed representation of the cross-sectional plane. Additionally, it can also recover detailedstress and strain fields based on inputs from a one-dimensional (1D) global analysis. Ituses the Variational Asymptotic Method15 (VAM) as its mathematical basis. VAM splitsa general 3D nonlinear elasticity problem for a beam-like structure into a two-dimensional(2D) linear cross-sectional analysis and a 1D nonlinear beam analysis by taking advantageof certain small parameters inherent to the structure (typically a/l and a/R, where a isa characteristic cross-sectional dimension, l is the wavelength of deformation and R is theradius of curvature/twist). VAM applies an asymptotic expansion of the energy functionalin terms of these small parameters (instead of the system of differential equations16,17,18,19),thereby making the modeling procedure more compact, less cumbersome, and variationallyconsistent. The development of VABS was first mentioned in Ref. 20. The cross-sectionalmodeling capability was later extended to include refinements such as transverse shear andeffects of initial curvature and twist21,22,23,24 and most recently updated in Ref. 25 andverified in Ref. 26.

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III. Flying Wing Model

These analyses used NATASHA, which is a geometrically exact beam analysis code to cal-culate the behavior as if it were a 1D beam. This type of analysis is suitable for studyingwings of high aspect ratio with reasonably continuous structures, and has the capabilityto calculate the non-linear equilibrium trim state, and then linearize about that state andperform an eigenvalue analysis to determine the stability of the equilibrium state.

A. Geometric Description of the Model

The configuration chosen for this study was inspired by the HORTEN flying wing model,13

but was altered significantly to change the character and flight speed of the flutter instabil-ity to a body-freedom flutter motion involving body pitching and plunging (“short-period”motion) and wing out-of-plane bending. The geometry of the model is realized by using1 beam to represent the main wing. A total of 40 elements were used to represent thewing, 38 flexible elements to represent the left and right wing, and two rigid elements torepresent the offset of the wing from the fuselage centerline. To eliminate whatever effectengine placement might have had on the body-freedom flutter characteristics of the model,the engines were placed at the root of the wing and aligned with the reference axis. Theresulting finite-element model is depicted in Fig. 1 and some relevant dimensions are givenin Table 1a.

−10 −8 −6 −4 −2 0 2 4 6 8 10−5

−4

−3

−2

−1

0 X X

Figure 1: Finite element model of the typical geometry for input into NATASHA. The axesare in units of meters, and the “X” marks indicate the location of the fuselage CG andtwo engines at their root location. See Table 1a for numerical details about the baselinegeometry.

B. Geometric Description of a Typical Section

The cross-section is defined by a NACA 0012 airfoil profile.. A very simple structure witha “I”-beam and skin was used to find the smallest possible stiffness results using VABS.A picture of the VABS model for the root section is shown in Fig. 2. Aluminum materialproperties were used in the VABS analysis, and a minimum skin and spar thickness of 0.127cm was used to recover the body-freedom flutter motion at a reasonably low flight speed.The low flutter speed was desired so that changes to the model that increase the flutter speedwould not drive it so high to violate the incompressible assumptions made in the aerodynamic

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model of NATASHA. The resulting mass per length, torsional and bending stiffness from thebaseline VABS analysis can be found in Table 1b. For each analysis, VABS was run for eachelement along the wing, so an accurate distribution of stiffness and inertial properties wasobtained. The thickness of the skin and “flange,” the horizontal sections of the spar, werevaried in this study.

Table 1: Geometric, structural, and inertial NATASHA inputs for the typical flying wingmodel. H refers to the angular momentum of the engine. Note that the fuselage inertiaswere set as functions of the fuselage mass µf .

(a) Overall Geometric Details

Parameter Value

Sweep (deg) 20

Dihedral (deg) 0

Initial Twist (deg/m) -0.2 /m

Wing Offset (m) 1.65

Span (m) 20

cr m 1.0

ct m 0.25

(b) Root section structural data.

Parameter Value

µ (kg/m) 9.761

GJ (N-m2) 4.24× 105

EI2 (N-m2) 3.84× 105

EI3 (N-m2) 2.46× 107

cref m 1

(c) Weight and inertia for the engine and fuselage.

CG Value Engine Fuselage

mg,(N) 51.445 150

Ixx (kg-m2) 0.29547 0.5 µf

Iyy (kg-m2) 0.29322 1.0 µf

Izz (kg-m2) 0.29547 1.0 µf

H (N-m-s) 5.24 N/A

chordwise offset (in) 0.0cr 0

vertical offset (in) 0.0cr 0

−25 −13 0 13 25 38 51 64 76

−6−4−2

0246

Figure 2: Very simple cross-sectional geometry used for VABS analysis to obtain baselinestiffnesses. The axes have units of centimeters.

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C. Engine/Fuselage Description

The engines and fuselage are represented by their mass and inertial properties. The fuselagemass and inertia is added at the reference node of the wing and the engines are added atnodes an equal distance left and right from the reference node. Table 1c shows the currentvalues for the weights and inertia of the engines and fuselage: this inertial information wastransformed appropriately to account for the fuselage offset and orientation of the wing withrespect to the engine. All cross-terms in the inertial matrix for both the engine and fuselage(such as Ixy) are assumed to be zero.

D. Aerodynamic Model

NATASHA uses a strip theory aerodynamic model which assumes that each element’s aero-dynamics are independent and do not interfere with each other, based on the unsteadytheory developed by Peters et. al.27 The aerodynamic coefficients were set to be constantover the span, and since the airfoil is symmetric cl0 = cm0 = 0 was assumed. The thin airfoiltheory value of clα = 2π was used and cmα = −0.08 was determined by using the onlineresource JavaFoil for a NACA0012 profile. The root and tip values of each coefficient arethen presented in Table 2.

Table 2: The aerodynamic coefficients used in the NATASHA model. The airfoil profile andcoefficients were held constant along the span.

clo clα clδ cdo cdα2 cdδ2 cmo cmα cmδ

0 2π 1 0.01 0 0 0.0 -0.08 0

E. Baseline Flutter Results and Eigenvalue Analysis

The flutter analysis of the baseline model resulted in a flutter speed of approximately 32m/s. The flutter mode shape was found to be a true body-freedom flutter motion, withbody pitching and plunging (“short-period” motion) and symmetric out-of-plane bending ofthe wings. The evolution of the lowest few eigenvalues are shown in Figure 3 in terms ofdamping and frequency (Hz). Upon visualization of each of the mode shapes, the eigenvalueswere qualitatively characterized by the type of motion observed, and this characterization canbe found in Table 3. Note that the titles associated with each mode in Table 3 describe the“dominant” motion of the mode; for instance, the mode called “short-period” is dominatedby rigid body pitching and plunging, but contains significant components of bending aswell. It is also worthwhile to note a physical difference between “structural-dynamics” and“flight-dynamics” modes: as the flight speed is reduced to zero, the frequency of all “flight-dynamics” modes should go to zero, while the frequency of all “structural-dynamics” modesgo to the in vacuo frequencies. Figure 3 seems to show a coalescence between the bendingand short-period modes. The stability of the bending mode begins to decrease at around 27m/s, before the two modes have coalesced significantly, but the frequencies of the two modesare very close to one another at the flutter speed.

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The short period and bending modes can be examined more closely by characterizing theeigenvectors in terms of the phase and magnitude of the various components. The short-period and bending modes were found to have significant components of rigid body plungingand pitching, and wing symmetric bending. Therefore, the eigenvector components in rootplunging velocity (V3)r, root pitch rate Ω1, root bending moment M2, and the tip verticalvelocity (V3)t were examined in terms of their magnitude and phase. The eigenvectors werescaled so that (V3)r had a magnitude of 1 m/s, and the phases were taken with respectto this component as well (so this component always has 1 magnitude and 0 phase). Themagnitude and phase characterizations are shown in Figure 4. Figure 4 shows one reason the“short-period” and “bending” titles were assigned to each mode: the magnitude ofM2 is moresignificant in the “bending” mode. However, both modes have significant components of M2.This figure also shows that for both the short period and first bending mode, the magnitudesof M2 and Ω1 are relatively constant over the flight speed, while the (V3)t magnitude increasesfor the short-period mode but decreases for the bending mode. This decrease of tip velocitymagnitude is related to the phases of (V3)t, (V3)r and M2. Figure 4 shows that the shortperiod mode has (V3)t lagging (V3)r but the bending mode has (V3)t leading (V3)r. Also, thereis a significant change in the phase of (M2)r with increasing flight speed for both modes, withthe phase increasing with the short-period and decreasing for the bending mode. The rootbending moment is nearly in phase with the root plunging velocity at the onset of instability.

25 26 27 28 29 30 31 32−0.5

0

0.5

1

1.5

Flight Speed (m/s)

Dam

ping

(H

z)

25 26 27 28 29 30 31 320

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quen

cy (

Hz)

Figure 3: Eigenvalue analysis results for the baseline model. Instability is found at the flightspeed of 32 m/s.

Table 3: Characterization of baseline eigenvalue results in terms of structural vibration andflight dynamical modes.

Frequency Range (Hz) Characterization

0.0 – 0.1 Phugoid and Yaw Stability (two modes)

0.5 – 0.6 Lateral Stability “Dutch Roll”

1.3 – 1.7 Symmetric Out-of-Plane (SOP) Bending / Short-period

1.9 – 2.1 SOP Bending / Short-period

3.0 – 3.5 Anti-symmetric OP Bending

5.6 – 6.0 2nd SOP Bending

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24 26 28 30 320.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Flight Speed (m/s)

Rel

ativ

e M

agni

tude

24 26 28 30 32−100

−80

−60

−40

−20

0

20

40

Flight Speed (m/s)

Rel

ativ

e P

hase

SP Root M

2 x 0.001

SP Tip V3

SP Root O1 x 1e+01

SOP Root M2 x 0.001

SOP Tip V3

SOP Root O1 x 1e+01

Figure 4: Short-period (SP) and out-of-plane bending (OPB) eigenvector characterizationin terms of magnitude and phase of root bending moment M2, tip vertical velocity V3, androot angular velocity O1. Eigenvectors were normalized so that root plunging velocity (V3)rmagnitude was 1 m/s and then phase-shifted so that (V3)r phase is zero.

F. “Typical” Flying Wing Model

One method that was selected to explore the inputs that affect the body freedom flutterwas to create a “typical” flying wing model using as few inputs as possible and attempt toobtain the same behavior. Therefore the information from Tables 1a – 2 were used, andthe structural model was simplified further by assuming the following variation of structuralinputs over the span.

c = c/cref µ = µ0c2

[T ]0 =

0 0 0

0 1/EI2 0

0 0 0

[T ] = [T ]0/c3

(4)

The resulting model approximates the low frequency behavior of the realistic flying wingmodel very well. The effect of the various aerodynamic coefficients was explored by changingeach coefficient to an alternate realistic value one by one. This process affected the trimvalues significantly but only clα and cmα had a significant effect on the flutter behavior.In general, trim conditions are difficult to obtain with a non-zero cl0 , but Figure 5 showsthe eigenvalues for the baseline model and for different values of cl0 , cd2α , cm0 , cmδ

; thisFigure shows there is no distinction between these different cases in terms of these lowesteigenvalues, or the flutter speed. Therefore, even though unrealistic trim values may beobtained without an appropriate cl0 , the BFF behavior is unaffected. Figure 6 shows thatclα does affect these dynamics, especially in the frequency of the short-period and bendingmodes and the damping of most modes. It may be significant that lowering clα pushes thefrequency of the relevant BFF modes apart and increases the flutter speed, again suggestingthat the similarity of the short-period and bending frequencies leads to the instability. Inany case, the “typical” model demonstrates very clearly that wing bending flexibility, massper length, fuselage inertial properties, and clα are the dominant factors at play with thistype of BFF. The “typical” model also shows that aerodynamic design can be conductedlargely without consideration to the body-freedom flutter (BFF) issue, because even when

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clα was adjusted significantly the BFF speed changed only slightly.

25 26 27 28 29 30 31 32−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Flight Speed (m/s)

Dam

pin

g (

Hz)

25 26 27 28 29 30 31 320

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quency (

Hz)

Figure 5: Low frequency eigenvalues for the “typical” flying wing model for the baselineand after modifying cl0 ,clδ , and all coefficients related to cd and cm. Curves are not labeledbecause they coincide exactly.

25 26 27 28 29 30 31 32 33 34−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Flight Speed (m/s)

Da

mp

ing

(H

z)

25 26 27 28 29 30 31 32 33 340

0.5

1

1.5

2

2.5

3

3.5

4

Flight Speed (m/s)

Fre

qu

en

cy (

Hz)

clα = 2π

clα = 6

clα = 5

Figure 6: Control settings for the “typical” flying wing model for the baseline and aftermodifying clα .

IV. Trade Studies on Body Freedom Flutter

Several trade studies were conducted to examine the effect on the flutter characteristics ofchanging various physical inputs of the HALE aircraft. These included the fuselage mass, thefuselage CG location and its pitching inertia, and the constitutive and inertial informationof the wing. The constitutive and inertial information of the wing was first varied by varyingthe structural geometry and finding the change of constitutive and inertial properties withVABS, and then specific elements of the mass and stiffness matrix were varied independentlyto show their effect alone.

A. Effect of Fuselage Properties on Flutter Speed

The fuselage properties that were found to impact the body freedom flutter behavior werethe total mass, the fuselage inertia, and the fuselage CG location.

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Figure 7 shows the results of the fuselage mass factor sweep. The fuselage mass con-tributed the most effective changes to flutter speed out of the factors listed above. Theincreased fuselage mass also increased the fuselage inertia, so some of the effects in thisFigure are due to the increased inertia of the fuselage. To separate the effect of the fuselagemass from the fuselage inertia, the fuselage inertia was itself varied as the total mass washeld constant. Figure 8 shows the results of the fuselage inertia factor sweep. The pitchinertia of the fuselage should have a large impact on the frequency of the short-period mode:one approximation for the short-period frequency is V/I0.5x and a similar dependency wasobserved for the flutter frequency. Here there is a discrepancy between Figs. 7 and 8: for eachthe fuselage inertia was increased, but only when the fuselage inertia was increased alonethe flutter speed increased. Comparison between these two figures reveal that the decreasedflutter speed in Fig. 7 is due to the effect of the increased mass.

Figure 9 shows the results of varying the CG location. This shows that the flutter speedis highly dependent upon the CG location.

Figure 10 shows the results of the CG factor sweep in along the z-axis. Like the fuselageinertia, the flutter speed was lowest for the moderate range, reflecting the fact that loweringthe fuselage had the affect of increasing the fuselage inertia when calculated at the wing. Adifferent flutter mode occurred whenever the fuselage was placed more than 1.6 m below thewing reference line that was associated with in-plane wing bending and torsion.

0 1 2 3 420

30

40

50

60

70

80

Fuselage Mass Factor

Flu

tter

Spe

ed, m

/s

0 1 2 3 40.5

1

1.5

2

2.5

3

3.5

4

Fuselage Mass Factor

Flu

tter

Fre

quen

cy, H

z

Figure 7: Flutter speed and frequency as a function of a multiplicative fuselage mass factor.

0 0.5 1 1.5 230

35

40

45

50

55

Fuselage Inertia Factor

Flu

tter

Spe

ed, m

/s

0 0.5 1 1.5 2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Fuselage Inertia Factor

Flu

tter

Fre

quen

cy, H

z

Figure 8: Flutter speed and frequency as a function of a multiplicative fuselage inertia factor.

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−1.4 −1.2 −1 −0.8 −0.6 −0.430

40

50

60

70

80

90

100

110

CG Location

Flu

tter

Spe

ed, m

/s

−1.4 −1.2 −1 −0.8 −0.6 −0.40

1

2

3

4

5

6

CG Location

Flu

tter

Fre

quen

cy, H

z

Figure 9: Flutter speed and frequency as a function of CG location, measured forward ofthe wing.

−2 −1.5 −1 −0.5 0 0.5 130

40

50

60

70

80

90

100

Fuselage CG "Z" Location

Flu

tter

Spe

ed, m

/s

−2 −1.5 −1 −0.5 0 0.5 10

2

4

6

8

10

12

Fuselage CG "Z" Location

Flu

tter

Fre

quen

cy, H

z

Figure 10: Flutter speed and frequency as a function of fuselage z-location.

B. Effect of Changing Structural Geometry

In order to demonstrate more directly the effect of changing structure, trade studies wereconducted which varied the skin or flange thickness and used VABS to calculate the cor-responding change in stiffness properties. Figure 11 shows the flutter speed and frequencyresults when varying the skin thickness in the VABS analysis, and Fig. 12 shows the flutterspeed and frequency results when varying the flange thickness. These figures show thatincreasing the structural geometry indeed had the effect of increasing the flutter speed, asexpected. The two structural geometry changes affected the constitutive properties in differ-ent ways, as shown in Fig. 13: the skin thickness increase affected many of the constitutiveproperties, while the flange thickness increase mainly affected bending stiffness and massper length. The effectiveness of the flutter increase in terms of increased mass and increasedbending stiffness is shown in Fig. 13. This figure shows that the flange thickness increase wasmore effective in increasing the flutter speed in terms of both mass and stiffness. The rightside of Fig. 14 shows that the additional constitutive property increases caused by increasedskin thickness actually had an adverse effect on the flutter speed.

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0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4532

33

34

35

36

37

38

39

Skin Thickness, cm

Flu

tter

Spe

ed, m

/s

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.451.8

1.85

1.9

1.95

2

Skin Thickness, cm

Flu

tter

Fre

quen

cy, H

z

Figure 11: Flutter speed as a function of increasing skin thickness.

0 0.2 0.4 0.6 0.8 1 1.2 1.432

33

34

35

36

37

38

39

40

Flange Thickness, cm

Flu

tter

Spe

ed, m

/s

0 0.2 0.4 0.6 0.8 1 1.2 1.41.8

1.85

1.9

1.95

2

Flange Thickness, cm

Flu

tter

Fre

quen

cy, H

z

Figure 12: Flutter speed as a function of increasing flange thickness.

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0 0.5 11

1.5

2

2.5

3

Geometry Increase

GJ

(N m

2 )

0 0.5 11

1.5

2

2.5

Geometry Increase

EI 2 (

N m

2 )

0 0.5 11

1.5

2

2.5

3

Geometry Increase

EI 3 (

N m

2 )

0 0.5 11

1.5

2

2.5

3

Geometry Increase

mu

(kg/

m)

0 0.5 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Geometry Increase

ξ 2, m

0 0.5 11

1.5

2

2.5

3

Geometry Increase

i 22, N

Skin ThicknessFlange Thickness

Figure 13: Change in constitutive properties as skin thickness and flange thickness were in-creased from their minimum (Geometry Increase = 0) to their maximum (Geometry Increase= 1). Skin thickness was varied from 0.127 to 0.381 cm, and flange thickness varied from0.127 to 1.016 cm.

5 10 15 20 25 3032

33

34

35

36

37

38

39

40

Mass Per Length (kg/m)

Flu

tter

Spe

ed (

m/s

)

3 4 5 6 7 8 9 10

x 105

32

33

34

35

36

37

38

39

40

Bending Stiffness (N m2)

Flu

tter

Spe

ed (

m/s

)

Skin ThicknessFlange Thickness

Figure 14: Flutter speed as a function of increasing skin and flange thickness as a functionof mass per length of the root section (left) or bending stiffness of the root section (right).

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C. Effect of Changing Individual Constitutive Properties

The increase in skin thickness significantly affected nearly all of the constitutive properties.The increase in flutter speed observed for increasing skin thickness was not as significantas would be expected as if the bending stiffness were increased alone. The effect of eachconstitutive property was isolated by changing each individually and observing the increasein flutter speed. It was found through these trade studies that many of the constitutiveproperties had a small or negligible effect on the flutter speed with the exception of bendingstiffness and mass per length. Figure 15 shows the results of the skin thickness sweep interms of flutter speed compared with the flutter speed results when only the bending stiffnesswas varied and when only the mass per length (µ) was varied. This Figure shows that whilethe increased bending stiffness raised the flutter speed, while the increased mass per lengthlowered the flutter speed.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4530

35

40

45

50

Flu

tter

Spe

ed, m

/s

Skin Thickness, cm0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Skin Thickness, cm

Flu

tter

Fre

quen

cy, H

z

All Properties VariedBending Stiffness Distribution Variedm Distribution Varied

Figure 15: Flutter speed as a function of increasing skin thickness, with each constitutiveproperty varied independently.

V. Eigenvalue and Eigenvector Analysis for Various Cases

The evolution of eigenvalues and eigenvectors as a function of flight speed for various caseswas explored to see if insight into the nature of this type of body freedom flutter could begained. The cases of maximum skin thickness, maximum flange thickness, low fuselage massand high fuselage inertia were selected as they all exhibited significant increase in flutterspeed.

A. Maximum Skin Thickness

The case of maximum skin thickness has drastically increased stiffness, mass and inertialproblems in comparison to the baseline. Figure 16 shows how the eigenvalues change as afunction of flight speed for this case. Figure 17 characterizes the short-period and bendingmodes in terms of magnitude and phase, with the eigenvectors normalized and phase-shiftedso that root plunging velocity has a magnitude of 1 m/s and a phase of 0. The qualitativecharacterization of each eigenvalue has not changed from those given in Table 3.

Figure 16 shows that the frequencies of the short-period and bending modes have indeed“coalesced” in the sense that they come to the same frequency at around 31 m/s, andafterwards the frequencies are very close between the two modes. Before this coalescence,

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the damping of the bending mode is increasing with flight speed, but after the coalescencethe bending mode damping decreases rapidly with respect to flight speed.

Figure 17 again shows that tip velocity magnitude increases with flight speed for theshort-period mode but decreases with flight speed for the bending mode. Interestingly, eachcomponent has similar magnitudes in the two modes near the speed where the frequencycoalesces. The magnitudes shown in Fig. 17 have trends that closely resemble the trendsfound in the baseline case (Fig. 4).

Figure 17 again shows many similarities to Fig. 4 in the phase results, with bendingmoment phase approaching zero at the onset of stability, tip velocity leading the bodyplunging velocity for the short-period mode and lagging the body plunging velocity for thebending mode. Again the characteristics of the two eigenvectors in terms of phase are similarnear the “coalescence” flight speed of 31 m/s.

26 28 30 32 34 36 38−0.5

0

0.5

1

1.5

2

Flight Speed (m/s)

Dam

ping

(H

z)

26 28 30 32 34 36 380

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quen

cy (

Hz)

Figure 16: Eigenvalue analysis results for the model with maximum skin thickness (0.381cm). Flutter speed is 38.2 m/s.

25 30 35 400

0.5

1

1.5

2

2.5

Flight Speed (m/s)

Rel

ativ

e M

agni

tude

25 30 35 40−100

−80

−60

−40

−20

0

20

40

Flight Speed (m/s)

Rel

ativ

e P

hase

SP Root M

2 x 0.001

SP Tip V3

SP Root O1 x 1e+01

SOP Root M2 x 0.001

SOP Tip V3

SOP Root O1 x 1e+01

Figure 17: Short-period (SP) and out-of-plane bending (OPB) eigenvector characterizationin terms of magnitude and phase of root bending moment M2, tip vertical velocity V3, androot angular velocity Ω1. Eigenvectors were normalized so that root plunging velocity V3magnitude was 1 m/s and zero phase. Results are shown for the maximum skin thickness(0.381 cm).

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B. Maximum Flange Thickness

The case of maximum flange thickness has increased bending stiffness and mass per length,but many of the other constitutive properties remain constant. Figure 18 shows how theeigenvalues change as a function of flight speed for this case. Figure 19 characterizes theshort-period and bending modes in terms of magnitude and phase, with the eigenvectorsnormalized and phase-shifted so that root plunging velocity has a magnitude of 1 m/s anda phase of 0.

Examining these three figures in comparison with Figs. 3, 4, 16, and 17, we begin to seea trend in the behavior of the eigenvalues and eigenvectors. The “coalescence” flight speedfor this case would be approximately 33 – 35 m/s, although the shift in the stability of thebending mode still occurs around 31 m/s. The cross-over of tip velocity magnitudes in Fig.19 coincides with the shift of stability of the bending mode, while the magnitudes of M2

and Ω1 approach each other near the coalescence flight speed. Again, the bending mode hasdecreasing tip velocity magnitudes with increasing flight speed while the short-period modehas increasing tip velocities with increasing flight speed. Fig. 19 also shows that for theshort-period the tip velocities lead the centerline velocities while the bending mode has aphase lag between the two velocities. Finally, the phase of the bending moment M2 is nearlyaligned with the plunging velocity at the onset of instability for the unstable mode.

26 28 30 32 34 36 38−0.5

0

0.5

1

1.5

2

Flight Speed (m/s)

Dam

ping

(H

z)

26 28 30 32 34 36 380

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quen

cy (

Hz)

Figure 18: Eigenvalue analysis results for the model with maximum flange thickness (1 cm).Flutter speed is 39.4 m/s.

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25 30 35 400

0.5

1

1.5

2

2.5

Flight Speed (m/s)

Rel

ativ

e M

agni

tude

25 30 35 40−100

−80

−60

−40

−20

0

20

40

Flight Speed (m/s)

Rel

ativ

e P

hase

SP Root M

2 x 0.001

SP Tip V3

SP Root O1 x 1e+01

SOP Root M2 x 0.001

SOP Tip V3

SOP Root O1 x 1e+01

Figure 19: Short-period (SP) and out-of-plane bending (OPB) eigenvector characterizationin terms of magnitude of root bending moment M2, tip vertical velocity V3, and root angularvelocity Ω1. Eigenvectors were normalized so that root plunging velocity V3 magnitude was1 m/s and phase was zero. Results are shown for the maximum flange thickness (1 cm).

C. Minimum Fuselage Mass

Figures 20 and 21 characterize the eigenvalue and eigenvector behavior over the flight speedsthat were analyzed. They show a slightly different picture of what is happening then whathappened with the nominal fuselage mass. Figure 20 shows the SP and OPB modes “coales-cence” of sorts, but the frequencies never meet each other, instead they begin to increase dras-tically. The lack of “coalescence” for this case is also partially due to the increased frequencyof the in vacuo bending mode. Figure 21 again shows the trend of increasing/decreasing ve-locity magnitudes for the SP/OPB modes with increasing flight speed over the flight speedsfrom 25-48 m/s. After this flight speed, however, the velocity magnitude trend reverses forthe bending mode and increases from 48-75 m/s. The speed where the velocities magnitudescross over one another still corresponds to a change in the stability of the bending mode interms of increasing flight speed. Figure 21 shows that for the bending mode the tip velocityphases from negative to positive around 48 m/s, at the same speed where the velocity trendchanges as mentioned above. The difference in stability and eigenvector trends between thiscase and the others is likely due to the much larger frequencies associated with the lowfuselage mass.

30 40 50 60 70−1

0

1

2

3

4

Flight Speed (m/s)

Dam

ping

(H

z)

30 40 50 60 700

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quen

cy (

Hz)

Figure 20: Eigenvalue analysis results for the model with small fuselage mass factor of 0.4.

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20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

4

Flight Speed (m/s)

Rel

ativ

e M

agni

tude

20 30 40 50 60 70 80−150

−100

−50

0

50

100

Flight Speed (m/s)

Rel

ativ

e P

hase

SP Root M

2 x 0.001

SP Tip V3

SP Root O1 x 1e+01

SOP Root M2 x 0.001

SOP Tip V3

SOP Root O1 x 1e+01

Figure 21: Short-period (SP) and out-of-plane bending (OPB) eigenvector characterizationin terms of magnitude of root bending moment M2, tip vertical velocity V3, and root angularvelocity Ω1. Eigenvectors were normalized so that root plunging velocity V3 magnitude was1 m/s. Results are shown for a small fuselage mass factor (0.4).

D. Maximum Fuselage Inertia

The eigenvalue and eigenvector characterization was repeated for the case with maximumfuselage inertia, and these characterizations are shown in Figs. 22 and 23. Figure 22 showsthat the “coalescence” does not occur as strongly as the previous cases, and also showsan unstable short-period mode as opposed to the unstable bending modes as found before.Figure 23 shows that the character of the eigenvectors has changed significantly, with thebending mode having much larger components of tip velocity and bending magnitude, andzero pitching velocity magnitudes over the whole flight speed regime. The bending modefrequency changed relatively little with increasing flight speed with this case, which maysuggest that the bending mode frequency changes are due in part to the pitching velocitycomponents. Both modes have increasing tip velocity magnitudes with increasing flightspeed. Figure 23 also shows that at the onset of instability the tip velocities are in phasewith the plunging velocities.

25 30 35 40 45 50−0.5

0

0.5

1

1.5

2

2.5

Flight Speed (m/s)

Dam

ping

(H

z)

25 30 35 40 45 500

1

2

3

4

5

6

7

Flight Speed (m/s)

Fre

quen

cy (

Hz)

Figure 22: Eigenvalue analysis results for the model with a large fuselage inertia factor 1.7.

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25 30 35 40 45 50 550

1

2

3

4

5

6

7

8

Flight Speed (m/s)

Rel

ativ

e M

agni

tude

25 30 35 40 45 50 55−100

−50

0

50

100

150

200

Flight Speed (m/s)

Rel

ativ

e P

hase

SP Root M

2 x 0.001

SP Tip V3 x 0.1

SP Root O1 x 1e+01

SOP Root M2 x 0.001

SOP Tip V3 x 0.1

Figure 23: Short-period (SP) and out-of-plane bending (OPB) eigenvector characterizationin terms of magnitude of root bending moment M2, tip vertical velocity V3, and root angularvelocity Ω1. Eigenvectors were normalized so that root plunging velocity V3 magnitude was1 m/s. Results are shown for a large fuselage inertia factor (1.7). Phase of O1 not shown forbending mode because for this case the magnitude of O1 for the bending mode is zero.

VI. Conclusions

The body-freedom flutter behavior of a HALE flying wing aircraft has been studied, andthe sensitivity of the behavior with respect to various parameters was explored. The BFFwas characterized as an interaction between the short-period and bending modes, so theevolution of these modes from the stall speed to the flutter speed for some selected caseswas examined. The structural properties that were found to impact this BFF behavior werethe bending stiffness and the mass per length. When the structural parameters are changed,the basic character of the modes was not changed, and flutter occurred in the bending modeand seemed to be associated with decreased tip velocity magnitudes, root bending momentsin phase with plunging velocity, and a “coalescence” of the short-period and bending modefrequencies. For the baseline and all of the cases with varied structure, the bending modebecomes unstable.

The basic character of the short-period and bending modes can be changed by alteringthe fuselage parameters, in some cases pushing the instability to a higher flight speed and insome cases causing the instability to occur at very low flight speed. The inverse dependencyof the flutter speed on the fuselage mass seems to reflect a dependence upon some sortof inertial coupling between the wing and fuselage motion, so that if the fuselage werenot present the BFF can not occur. Increasing the pitching inertia of the fuselage hadthe effect of removing the pitching velocities from the bending mode entirely, and for themaximum fuselage inertia factor the short-period mode experienced the instability (insteadof the bending mode). It might be said that the lack of angular velocity in the bending modecaused its frequency to change less with increasing flight speed, and therefore stabilized themode by avoiding “coalescence” between the bending and short-period modes. This reflectsthe fact that angular velocity is a necessary component to the body-freedom flutter behaviorof this model.

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Documents

[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