flutter and directional stability of aircraft

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Flutter and Directional Stability of Aircraft withWing-Tip Fins: Conflicts and Compromises

Matthew P. Snyder

European Office of Aerospace Research and Development,Reyslip, England HA4 7HB, United Kingdom

and

Terrence A. Weisshaar

Purdue University, West Lafayette, Indiana, 47907

DOI:10.2514/1.C031978

An alternative to vertical tails, wing-tip fins, a type of winglet, provide yaw stability and control and some

operational and maintenance advantages. Winglets reduce wing-induced drag but may reduce flutter speed,

requiring wing stiffening and a structural weight penalty. Whereas the number of successful aircraft with winglets

suggests that many flutter studies have been done, few of these studies appear in the archival literature. This paper

investigates the effect of tip-fin surface geometry on flutter as well as on directional stability. The study includes the

effects oftip-fincant anglewithrespectto theprimarysurfaceand finsizewithrespectto thewingitself.Twoidealized

models, a simple four degree-of-freedom RayleighRitz model and a high-fidelity finite element model, illustrate

special features of tip-fin aeroelasticity. Results indicate that the interaction between tip fins and the wing flexibilityusually leads to reduced flutter speeds but can also create phenomena such as mode switching that actually increases

flutter speed if the fin is large enough. Classical wing-surface lift ineffectiveness due to aeroelastic interaction also

reduces directional stability.

Nomenclature

a = wing displacement generalized coordinatesCl = rolling moment derivative contribution due to yaw

angleCn = yaw-moment derivativeCn = yaw-moment derivative contribution due to sideslipCnf

= yaw-moment derivative contribution from thefuselage due to sideslip

Clw

= yaw- moment derivative contribution from thewingdue to sideslip

Cnv = yaw-moment derivative contribution from thevertical tail due to sideslip

Cna = yaw-moment derivative contribution due to ailerondeflection

Cnr = yaw-moment derivative contribution due to rudderdeflection

Cy = side-force derivative contribution due to yaw anglec = chord lengthEI = bending stiffness of wingGJ = torsional stiffness of wingKij = structural stiffness matrixL = wing lengthl = roll moment

Lf = fin lengthlv = moment arm for calculating yaw-momentLiy = lift per unit spanm = wing mass per unit lengthMij = mass matrix

n = yaw-momentp = roll momentq = dynamic pressureQi = aerodynamic matrixr = yaw rateT = kinetic energyU = strain energywx;y;t = surface displacementwSCy; t = shear center displacement

x = distance in chordwise directionyw = distance in span direction (RayleighRitz flutter

model)y = side-force (directional stability section)z = distance out-of-plane (zcoordinate axis) = sideslip (yaw) angleiy = shape functiona = aileron deflection angle (rad)r = rudder deflection angle (rad)waero = virtual work

yL

= nondimensional position in span directiony; t = wing twist = wing sweep anglex; y = wing mass per unit areai

= real part of eigenvalue = imaginary part of eigenvaluei = system eigenvalues

I. Introduction

T HE primary purpose of this paper is to determine how thepresence of a large wing-tip fin, an oversized version of current-day winglets, affects flutter speed. A second objective is to provideinformation about static aeroelastic behavior that can change thedirectional stability of tip-mounted vertical surfaces. To do so, a briefhistory of winglet flutter is first examined. Results are then presentedfor a generalconfiguration using a simplemodeland a higher-fidelityflutter model.

Several current commercial aircraft use winglets to decreaseinduced drag at cruise speeds. Because these aircraft are certifiedthrough an extensive analysis, test, and certification process, flutter is

Presented as Paper 2012-1455 at the 53rd AIAA Structures, StructuralDynamics and Materials Conference, Honolulu, HI, 2326 April 2012;received 30 May 2012; revision received 10 August 2012; accepted forpublication 14 August 2012; publishedonline 29 January 2013.This materialis declared a work of the U.S. Government and is not subject to copyrightprotection in theUnited States. Copiesof this paper maybe made forpersonalor internal use, on conditionthat the copierpay the$10.00 per-copy fee to theCopyrightClearance Center,Inc., 222 RosewoodDrive,Danvers, MA 01923;include the code 1542-3868/13 and $10.00 in correspondence with the CCC.

*Program Manager, Air Force Office of Scientific Research. SeniorMember AIAA.Professor Emeritus, School of Aeronautics and Astronautics, 701 W.

Stadium Ave. Fellow AIAA

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JOURNAL OF AIRCRAFTVol. 50, No. 2, MarchApril 2013
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not a problem with these aircraft. On the other hand, very littlearchival literature on the effect of winglets on flutter exists.

Recent conceptual designs feature relatively large winglets, somewith a variable winglet cant angle with respect to the wing plane. In2007, the U.S. Air Force announced the Future Responsive Access toSpace Technologies (FAST) program [1]. If successful, the programwill produce a subscale X-aircraft ground test and flight test. Two

FASTvehicle concepts were proposed.One of these concepts,shownin Fig.1, features a wing tip-mounted fin design feature to providedirectional control [2].

Winglet/tip-fin designs such as those shown in Fig.2are not new.They provide some advantages over conventional centerline taildesigns, including easy access to engines and increased payload on-load/off-load options [4]. Tip-mounted fins also provide directionalstability and control. However, design uncertainties, in particular thepossibility of reduced wing flutter speeds due to fin/rudder modalcoupling, raise seriousquestions. It is wellknownthat winglet additionmay reduce flutter speed. A recent article in Aviation Week, discussedthe disappearance of the winglets shown on the original Boeing KC-46A tanker:the winglets did not earn theirway onto the airplane [5].

Winglet interest and design emerged from a desire to reduce

aircraft fuel consumption through increased aerodynamic efficiency.However, this increased efficiency comes with a trade-off, reducedflutter speed. Doggett and Farmer [6] studied this decrease in flutterspeed using a flat plate model with attached wing-tip fins. They usedboth light (0.3% of wing weight) and heavy (1.5% of wing weight)fins. Tests demonstrated that the light fin reduced the flutter dynamicpressure by 3%, whereas the heavy fin reduced the flutter dynamicpressure by 12%. These changes are due primarily to two effects:1) changes in the aerodynamic loads and 2) changes in the modeshapes and natural frequencies. Their experimental results correlatedwell with analytical solutions obtained using a finite element solver.

Shollenberger et al.[7] reported on flighttestresults andlow-speedwind-tunnel flutter model tests of winglets on the DC-10 airplane.Modal coupling between fin and higher-order wing modesmoderately reduced the flutter speed. As a result, ballast was added

to each wing tip to prevent the onset of flutter during testing.Kehoe[8,9] discussesa series of flighttests for winglets developed

for the KC-135A tanker aircraft. The winglet cant angle and inci-dence angle were adjustable on the ground so that the effects of cantangle on flutter speed could be studied.

Ruhlin et al. [10] and Bhatia et al. [11] investigated the effects ofadding a winglet on flutter speed. The study by Ruhlin et al. [10]showed a 7%decrease in flutter speed, the majority of which was dueto the effect of added wing-tip mass. The study by Ruhlin et al. [ 10]showed a flutter speed reduction up to 19%.

Several articlesdiscuss winglet aerodynamiceffectson performance[1216]. Researchers at Bristol University optimized variable cantangle winglets, called morphlets, to maximize a specific range [17,18].The researchers identified three cant angles that improved efficiencythroughout the flight envelope. While the addition of morphlets addsweight, reduced induced drag and increased range provide benefits.

Van Dam [19] showed that retrofitting an aircraft with wingletssignificantly improves crosswind takeoff and landing capabilities.

However, winglets were shown to have a detrimental effect on lateralstability and control because they tend to exacerbate adverse yawcaused by aileron deflection. Changing the winglet cant angle miti-gates this problem.

Roskam provided an overview and calculation method for thedetermination of lateral stability and control derivatives for centerlinetail vehicles [20]. Rahman and Whidborne analyzed a blended wingbody configuration with winglet rudders intended for lateral control[21]. Their study showed that these winglets did not provide the

necessary handling characteristics. As a result, they added centerlinevertical surfaces with rudders. Although the design features of theirtip-mounted surfaces are not fully documented, the winglets used ontheir vehicle appear smaller than those proposed for the FASTvehicle.

Bourdin et al. [22] examined aircraft control with variable cantangle winglets. His study allowed the cant angle to change duringflight. Testing in a wind tunnel measured forces and moments due towinglet deflection that were used to predict pitch, roll, and yawchanges. Their work demonstrated that winglets alone could notcontrol flight. In particular, a level turn was only possible for aspecific turn radius.

Several airplanes have used out-of-plane configuration changes tomaintain performance overtheir operational envelope. These includethe MiG-105 aerospaceplane, with variable cant angle wing tips, andthe XB-70 Valkyrie, whose wingtips rotated down at supersonicspeeds. The MiG-105 wing tips were used as vertical stabilizersduring high-speed flight and extended to a horizontal, in-plane, low-drag configuration in the subsonic regime. The outer wing panels ofthe XB-70 folded downward, doubling the size of the effectivevertical tail area, providing additional directional stability thatreduces required trim, thus reducing drag [23].

While a designer sees favorable changes in system performance,an aeroelastician sees potential problems that may reduce perfor-mance due to weight changes required to prevent flutter or staticaeroelastic effects such as surface-lifting effectiveness. These typesof aeroelastic problems were addressed during the Defense Advan-ced Research Projects Agencys Morphing Aircraft Structures(MAS) program, which ran from December 2002 to June 2006. The

objective of the MAS program was the design and fabrication ofeffective combinations of integrated wing skins, actuators andmechanisms, structures, and flight controls to achieve the anticipateddiverse, conflicting vehicle mission capabilities via wing shapechange[24].

The program led to two full-scale designs, one produced byLockheed Martin, the other by NextGen Aeronautics. Both designswere tested successfully in the Transonic Dynamics Tunnel (TDT) atNASA Langley Research Center at speeds up to Mach 0.92.Lockheed Martins designconcept, reconfigured by folding the innerwing against the fuselage, resulting in a 22% mission radiusadvantage over the best representative from a series of conventionalairvehicles[25].Tomove the wing, itis unlocked, and thena seriesofactuators power it into the desired configuration. However, this

action leads to large changes in bending stiffness, aerodynamicfeatures, and vibration modes. This resulted in a series of studiesconducted to investigate these effects on flutter of the folding wing[2631].

One notable result from these studies is that the folding wing ismost susceptible to flutter when the hinge fold angle is close to zero,wing fully extended. The flutter studies encompassed a range of foldangles, and for each angle the wing was held in a locked position. Asthe wing folds and hence shortens, the flutter velocity increasesrapidly before leveling off.

These few studies lead us to believe that it is important fordesigners to understand and document the aeroelastic features ofvariable cant angle tip-fins. The following sections describe the twomodels used for our study. Results are presented to show specialfeatures such as mode switching associated with large tip-fins. These

features have not previously been observed or documented.

Fig. 1 Conceptual sketch of proposed hypersonic vehicle with tip-mounted fins with large cant angle.

Information available at http://www.buran.ru/htm/molniya3.htm.

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II. Model Development

To identify aeroelastic issues relatedto the addition of a tip-fin to awing design, one must first select the key design parameters likely toinfluence the vehicle design in general and aeroelastic behavior inparticular. A listing of these parameters is given in Table1.

There are so many design parameters that, when taken incombination, the number of combinations of parameters that couldbeconsideredis extremelylarge. Thechallenge is to confine thestudytothose considered to be most essential in the early stages of design. Inour case, we chose those parameters and flight conditions shown inbold; these parameters are more likely to be included in preliminarydesign studies. The tip-fin size,both geometric and inertial, relative to

thewingplanformis very importantbecause it is also importantto thestability and control sizing. Wing sweep is also important.

The basic configuration chosen for this study is shown in Fig.3; itis a constant chord, sweptback wing with an unswept aerodynamicsurface mounted at the tip. This planform captures the essentialfeatures of the wing/fin interaction. The baseline wing parametersused forour study arethe well-known Golandwingplanform[32,33].We chose this wing because of its extensive documentation.

Two analytical models were developed. The first and simplestmodel is a RayleighRitz beam model, in which the wing and finstructures are both idealized as uniform beams with bending andtorsional degrees of freedom. A quasi-steady aerodynamic repre-sentation for the unsteady forces and moments provides an airspeed-dependentaerodynamic stiffness.This model restricts the tip-fin to be

mounted in a vertical position only.Wing deformation includes two assumed displacement modes,

one for bending and the other for torsion. The displacement at anypoint on thewing surface,wx;y;t, is composed of upward bendingat the wing shear center, wSC, and twist about the shear center,. Interms of thewingspanwisecoordinate,y, measured outward from thecantilevered wing root,and the chordwise coordinate, x, measured aftfrom the wing elastic axis, the displacement of any point on the wingisw1:

w1 wwingx; yw; t wSCyw; txyw; t (1)

The tip-fin motion is composed of two different types of motion. Thefirst is denoted as w2 and is due to out-of-plane fin bending and

torsion, a function of the fin spanwise coordinate, yf, and the finchordwise coordinate,xf.

w2 wfxf; yf; t wfin;SC xff

dwwingSC L

dycos wingL sin

yf (2)

The second contribution to tip-fin motion is an in-plane motion,denoted asw3, due to the wing tip upward movement. This is writtenas

w3 zf wwing;SCLxftipiy yftipix (3)

3 tip wingL cos wwing;SC

yL sin (4)

Polynomials were chosen as assumed displacement modes for theRayleighRitz analysis. These polynomials are exact solutions forthebending andtorsional shapesfor a beam loadedby a uniform load.

wSC a11yw a1

1

244

1

63

1

42;

yw

L

1 wing a22yw a2

1

22

wf;SC a33yT a3

1

244T

1

63T

1

42T

;

T

yT

LT

2 f a44yT a4

T

1

22T

5

The mass, stiffness, and aerodynamic force matrices are calculatedusing expressions for the wing/tip-fin kinetic energy, the wing/tip-finstrain energy, and the virtual work done by the quasi-steadyaerodynamic forces acting on the wing and tip-fin. The result, usingLagranges equations and assuming harmonic motion, is a fourth-

order eigenvalue problem of the form

2ml3Mijfaig EI

lKijfaigQijfaig f0g (6)

Table 1 Parameters affecting tip-fin aeroelasticity

Geometric Structural Aerodynamic Actuators

Surface sizes Hinge location Flight in 3 speedregimes:

Power

Element taper Hinge stiffness Subsonic Size:Surface sweep Inertia

characteristics:Supersonic Weight

Controlsurface size C.G. location Hypersonic VolumeControl surfacecant angle

Moments ofInertia

Accurate steadyand

Number

Rudder size unsteady loads

prediction

Distribution

Wing/fuselagelocation

Fig. 2 Various vertical tail designs: single centerline-mounted, dual fuselage-mounted, and wing-tip vertical fins [3].

Fig. 3 Planform geometry and dimensions of wing-tip mountedaeroelastic model.

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with

faitg faigeit (7)

The structural stiffness Ks and aerodynamic matrices Qi arecombined into a single matrixKij KijQij. Equation (6) iswritten as

2Mij Kijfaig f0g (8)

where

2

ml4

EI2 (9)

The eigenvalues, i, of Eq. (8) are functions of dynamic pressureand are either real numbers or complex conjugates of the formi i ii. Flutter is characterized by frequency merging withthe appearance of complex conjugate eigenvalues with non-zerovalues of i; divergence is signaled by a positive real eigenvalue

at zero frequency. Because of the quasi-steady nature of theaerodynamic forces, the eigenvalues will be real at speeds below theflutter speed.

Thesecond model used fortip-finflutter analysis is a finiteelementmodel developed using the Automated Structural OptimizationSystem(ASTROS) code [34]. This wing model, shown in Fig. 4,usesa single beam spar with beam elements splined to aerodynamicpanels to include a more accurate doublet lattice unsteady aero-dynamic model. The wing mass is provided by a series of con-centratedmasselements whose distance fromthe spar canbe adjustedto control the chordwise center of gravity (c.g.) position. The tip-finmass and stiffness parameters are identical to the wing; tip-fin lengthand cant angle are parameters for this model.

The geometry and material properties for the finite element modelwere identical to the RayleighRitz, wing model, but a doublet

lattice, unsteady aerodynamics model replaced the simpler quasi-steady aerodynamic representation, whereas the assumed structuraldeflection modes were replaced by a finite element structural model.

III. Flutter Results

Figure 5 shows typical flutter results from the RayleighRitzquasi-steady model with a 90 deg cant angle tip-fin length, Lf, halfthe wing semi-span,L. The set of black lines plot the real parts of thesystem eigenvalues for the wing without a fin. Flutter is indicated bythecoalescence of thewing-bending andtorsion modes.When thefinis added to the wing, four eigenvalues appear and there are twodifferent frequency coalescence curves. One of these curves willcoalesce at a lower flutter speed, depending on the fin size relative tothe wing.

The lower set of curves, blue and green, show flutter onset drivenprimarily by wing bending torsion interaction. This motion ismodified by thetip-fin. The secondset of curves, theupper setof blue

and red lines, coalesce at a higher airspeed; this interaction isprimarily between the tip-fin bending and torsion motion modifiedby

wing-tip motion at the tip-fin base connection. The relative positionsof these two coalescence points change with fin length. In this case,the flutter speed is reduced by the addition of the tip-fin.

Figure6shows flutter results from the two models showing howflutter speed changes as tip-fin length increases when the tip-fin isvertical. The results shown in Fig. 6 are also normalized to the speedspredicted by each model for the wing without the tip-fin. TheASTROS model always predicts higher flutter speeds than the quasi-steady model.

The ASTROS model indicates that flutter speed is reduced wheneven a small fin is added to thewing, whereas the quasi-steady modelshows a slight increase in flutter speed for small fins. Both modelsindicate that as the fin length increases, the flutter speed is reduced toless than half of the no-finvalue. These results are consistent with the

previously cited studies in [5], [9],and[10] whenthe fin isverysmall.Both modelsindicate that there aretwo flutter modes andindicate thatthere is a flutter mode switchingphenomenonnearLfL 0.5 whenthe tip-fin becomes large enough to drive the flutter instability.

These results show that, when the fin length exceeds 40% of thewing span, the flutter speed increases rapidly as tip length increases,but then declines as the result of a mode switching phenomenon. Toexplain the flutter mode switching, consider Fig.7, in which the fourin vacuo natural frequencies are plotted as a function of tip-fin lengthto wing semi-span ratio. As fin length increases, the two frequenciesdominated by wing bending and torsion approach each other. They

Fig. 4 ASTROS finite element model showing the wing and the tip-finwith its cant angle defined.

Fig. 5 Comparison of frequency merging, wing without tip-fin, wingwith tip-fin.

Fig. 6 Flutter speed change as a function of tip-fin length increase.

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then intersect near a point where the tip-fin length isabout 45% of the

wing semi-span. This is the point where the flutter speed begins toincrease again in Fig. 6. Before this increase theflutter mode is drivenby wing bending and torsion oscillations. As the tip-fin lengthincreases, the wing dominated natural frequencies separate while thetip-fin dominatedin vacuofrequencies are driven closer together. Asthe tip-fin becomes larger its motion begins to drive flutter, and theflutter speed is again reduced.

The wing-bending stiffness is, in part, a function of the load factoron the wing. A stronger structure is generally a stiffer structure.Although both the wing-bending stiffness and the torsional stiffnesswill be affected bythe size of the loadfactor imposed on the wing, weinvestigated the effects of changing only the wing bending stiffnesson flutter speed while the torsional stiffness is unchanged. Figure8shows plots of nondimensionalized flutter speed as a function of thetip-fin length to wing semi-span ratio for three different bendingstiffnesses. The figure on the left is generated using the RayleighRitz model, whereas the figure on the right is generated using theASTROS model. In each figure, the nondimensionalization is withrespect to thewingflutter speed without a tip-fin. These flutter speedsare different in each figure because the flutter models are different.

Both analyses show that decreased wing stiffnesstends tomovethecusp associated with mode switching to the right. In addition, theminimum flutter speed is also reduced. As before, these changes can

be traced back to the changes in spacing between the in vacuofrequencies.

Winglets are seldom mounted vertically with respect to the wingplanform. Figure9plots flutter speed as a function of the tip-fin cantangle. The ASTROS model was used for this study. In all cases, thewing semi-span is held constant while the tip-fin lengthchanges. Thereference flutter speed is that for a wing with no fin. At zero cantangle, the tip-fin is merely an extension of the wing tip so thatwings with the longer fins have a larger aspect ratio. As a result, the

addition of even a small tip-fin decreases the flutter speed for the zerodegree cant angle when the tip-fin merely serves as a wing-tipextension.

For a small fin, the flutter speed decreases with increasing cantangle. The longer tip-fins encounter mode switching phenomena asthe fin cant angle increases. Note that for a longer tip-fin, modeswitching occurs at a smaller cant angle. After mode switching, theflutter speed first increases but then decreases.

The two different models used to compute the likely effects ofmounting a tip-fin on a flexible wing show that there is a differencebetween theflutter behavior of large tip-fins andthat of small tip-fins.The small tip-fin flutter is driven primarily by the interaction of wingbending and torsion modes, modes whose in vacuo frequencies areaffected by the inertia features of the tip-fin. Flutter is affected to alesser extent by the aerodynamic forces on the tip-fin. As the size of

the fin increases, fin aerodynamics and fin flexibility play a moreprominent role in flutter. Finally, for larger fins, flutter modes aredominated by fin dynamics and aerodynamics. Fin cant angle alsoplays an important role in flutter. Very small cant angles drive theflutter speed down until a mode-switching phenomenon introducesfin driven flutter that increases the flutter speed.

Unlike drag reduction objectives that drive the design of winglets,the design of tip-fins for FAST-type vehicles is driven by directionalstability requirements. These requirements may produce relativelylarge fins. In this case aeroelastic lift effectiveness may become aproblem. Lift effectiveness refers to the condition in which the liftingsurface distortion reduces (or increases) the amount of lift producedper unit angle of attack. Lift effectiveness reduction is a problemwith moderately swept surfaces. The next section explores the lifteffectiveness of tip-fin configurations. These fins are modeled usingASTROS.

IV. Static AeroelasticityDirectional StabilityDerivatives

The previous section discussed the effects of tip-fin size on wingflutter. It showed that the addition of tip-fins to a clean wing canlead to reduced flutter speeds. Flutter is a major structural design

Fig. 8 Bending stiffness influence on flutter speed, a) RayleighRitz model and b) ASTROS model.

Fig. 7 Wing and fin natural frequencies decline as tip-fin lengthincreases.

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constraint. It isfair toask then, how large the tip-finsmustbe for theirprimary purpose of generating vehicle stability and how will staticaeroelastic effects intrude on the tip-fin design activity. No designactivity exists in a vacuum; design trade-offs appear and requireresolution. And so it is with the design of a wing with a tip-fin.

This section examines theimportance of staticaeroelasticity on theeffectiveness of these tip-fins when they are used to provide aircraftdirectional or yaw stability. Fin size and sweep angle are theparameters that drive the study of yaw stability.

Aircraft aerodynamic forces and moments depend on the vehicleorientation with respect to the flight trajectory. The sideslip angle,the primary parameter in directionalstability, describes therotation ofthe aircraft centerline with respect to the relative wind. The sideslipangle is the directional angle of attack of the airplane. Vertical tailsand tip-mounted fins generate side-force and a yaw-moment. Thissection focuses on tip-mounted fins that replace the vertical tail and

the associated directional stability derivatives; this overview does notconsider the control effects associated with a rudder.

Directional stability, also called weathercock stability, is thetendency of an aircraft to return to its equilibrium state whendisturbed in yaw. The fuselage forward of the aircraft c.g. produces aside-force, due to yaw, which the fuselage and vertical tail surface aftof the c.g. must counteract. The purpose of vertical tails and tip-mounted fins is to generate this restoring side-force and yaw-moment, either through intentional rudder deflection or sideslip.Sideslip angle is a yawing rotation that places the tail surface at anangle of attack and produces a restoring moment. The size of thisyaw-moment depends on surface size, placement on the aircraft,airspeed and altitude. Yaw-moment also depends on surface struc-tural flexibility.

Lateral stabilityrefers to theability of an aircraft to returnto a levelflight condition after it has been perturbed in roll. Designers builddihedral into wings to increase lateral stability. Kermode points outthat designers cannot separate coupling effects introduced by thevertical tail on lateralstability and directional stability[35]. Whitfordcomments that designing fins only for weathercock stability wouldlead to smaller vertical stabilizer sizes [36]. This section will containdiscussion andresults related to theeffects of aeroelasticity on severalaspects of vertical tail effectiveness.

It is standard convention to express flight vehicle aerodynamicforces and moments in terms of nondimensional force and momentcoefficients. These force and moment coefficients have the generallyaccepted nomenclature indicated in Fig.10([37]). This figure showsan aircraft with a positive sideslip or yaw angle, denoted as ,measured positive counter-clockwise. If the aircraft is stable, sideslipwill produce a restoring moment clockwise about a vertical axis.For flight mechanics computations, the positive direction of thevertical z-axis is directed downward so that this is a positive moment;

the yawing moment and the yawing moment coefficient for a stableaircraft are positive.

Standard convention defines stability and control derivatives withrespect to reference wing areas and characteristic lengths, usually awing span or chord length. The derivative magnitude is a linearsuperposition of the contributions from vehicle components such asthe wing and tail and due to the deployment of elements such as arudder.

For example, the yawing moment derivative Cn measures theability of the configuration to remain in a stable static and dynamicstate and to provide yaw stability. This derivative is defined as

N qSbCn (10)

hereNis the dimensional yawing moment, S is the reference wingarea,b is the wing span, and

Cn Cn0 Cn Cna a Cnr r (11)

Cn0 isthe value ofCnwhen a r 0. The terms aredue totherotation of the ailerons or rudder to produce roll duringflight. Theyawing moment produced by the sideslip angle, , is a sum ofcontributions from thewing, fuselage,and vertical tail(s). This term iswritten as

Cn Cnw Cnf Cnv (12)

The first term in Eq. (12) is the contribution from the wing, usuallynegligible except at large angles of attack. The second term is due tothe fuselage and, for most aircraft, this contribution is negative [38]

and diminishes stability. The third term, contribution from thevertical tail, determines yaw stability.

ASTROS provides the ability to calculate lateral and directionalstability derivatives: the yaw rate derivatives:Cyr ,Clr , Cnr ; the angleof sideslip derivatives: Cy, Cl, Cn; the roll rate derivatives: Cyp ,Clp ,Cnp . The subscripts refer to: y side force,l roll moment,n yaw rate, r yaw rate, sideslip angle, and proll moment.

However, of these nine derivatives, a conventional vertical tailprovides major contributions only toCn,Cy,Cnr ,Cyr [39]. The rollderivatives, Cnr ,Cyr , calculated with respect to the roll axis, describethe effect of the tip-fin on the yaw-moment and side-force resultingfrom roll, respectively.Cnr must be negative to provide stability; thetypical range of values for this derivative is between 0.1and 0.4[5]. Whether the verticaltailis onthe fuselageor ismovedto the wingtip, these derivatives will not change because the tip-fin moments donotchange with respect to theroll axis. As a result, this overviewdoesnot include a discussion of these derivatives.

Fig. 9 Flutter speed vs. tip-fin cant angle for four fin lengths.

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The yaw stability derivative,Cn, depends on the distance betweenthe aircraft c.g. and thevertical tail aerodynamic center, shown as thedistancelv in Fig.11. Note that Eq. (10) showed yaw-moment as afunction of wing span, b. This is a standard way to present theequation; however, when the yaw-moment is calculated,b is replaced

by the moment arm length for eachcomponent that contributes to theyaw-moment. Here, the moment arm for the vertical tail is shownbecause this study will address the stability provided by the tail. Oneway to increase this moment arm is to sweep the wing. When thevertical stabilizer is movedfrom thecenterline, lateraland directionalstability is maintained by adjusting fin parameters such as the area ofthe fin surface, fin sweep angle and fin cant angle.

With the coordinate system shown in Fig.10 the restoring yaw-moment and associated directional stability derivative are positivebecause thez-axis is directed downward. However, ASTROS uses adifferent coordinate system in which the z-axis is directed upwardandthe x-axis is positive aftso that thesign of therestoring momentisnegative. Figure12shows these differences.

If ASTROS calculations give negative Cn the air vehicle haspositive yawstability. The side-force derivative is notaffected by thiscoordinate system change.

Aeroelastic deformation has an effect on stability derivatives. Lifteffectivenessis defined as the lift produced by a flexible wing, at afixed angle of attack, divided by the lift produced by an identical, butrigid, wing at the same angle of attack. For this model, wing twistingincreases lift, but wing bending, when combined with wing

sweepback, decreases lift, leading to differences between rigid andflexible wings.A swept wing with twist and bending deformation has its local

angle of attack changed by an amount equal to cos dwdy

sin .Here, is thetwist angle,w is the upward deflection, y is thespanwisecoordinate, and is the wing sweep angle (see Fig.3). If the wing isunswept, bending deformation has no effect on the lift distribution;for sweptback wings the angle of attack is reduced.

To illustrate this aeroelastic effect, consider the wing-liftdistribution shown in Fig.13. This 35 deg sweptback wing-lift dis-tribution isshownfor both a rigid and a flexiblewing. Bothwings aretrimmed to the same total lift, such that the areas under the curves areidentical, although the trim angle of attack differs. The effect offlexibility is to move the spanwise center of pressure inboard, nearerthe wing root. While this is a desirable effect for a wing (the bending

Fig. 11 Vertical tail relationship to aircraft center of gravity.

Fig. 12 Difference between the classical flight mechanics stability axissystem and ASTROS axis system.

Fig. 10 Flight Mechanics terminology for yaw moments.

Weisshaar, T. A. AAE556Aeroelasticity Class Notes. PurdueUniversity, West Lafayette, 2010.

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moment is reduced), it is not a favorable effect for a vertical tail orhorizontal stabilizer because it reduces the distance between thesurface aerodynamic center and the aircraft c.g. In addition, the angleof attack for the flexible surface is larger so that it can generate morelift. The surface is less lift effective than the rigid surface.

An unswept tip-mounted vertical (cant angle 90 deg) stabi-lizer will have increased lift effectiveness because wing twist adds lift

and bending/sweep effects are not present. Figure14illustrates thisfor the Goland unswept wing model, for which thewing semi-span is17 ft. This figure shows ASTROS generated results for two sideslipderivatives, Cn, Cy, plotted against tip-fin size for a wing/tip-finoperating atM 0.8 at two different altitudes, 15,000 feet and30,000 feet. The blue line represents the rigid derivative while thedashed red lines show the flexible derivatives at two differentaltitudes. In this figure, increasing stability is indicated by larger(downward) negative numbers.

The horizontal axis shows nondimensional fin length, the finlengthis divided by theoriginalGoland wing lengthof 20 ft. Both therigid and flexible derivatives show that yaw stability increaseswith increasing fin length. The effects of aeroelasticity are greater atlower altitudes where air density is greater, leading to higherdynamic pressures. While not shown here, increasing or decreasing

the wing length changes the relative position of the curves to oneanother.

In addition tolift effectivenessconcernsdue to finlength, changingthe wing-and-fin-sweep angle also modifies the lift effectiveness ofthe surface. Figures1517show the results of an investigation ofsweep effects. Figure 15 shows the effect of sweep angle on fineffectiveness. While the analysis behind the figure considers twocases, 1) only the fin is swept and 2) both the wing and fin are swept,

Fig.15compares the rigid and flexible derivatives only for the firstcase. This plot shows resultsfor twofin lengths, a small, short fin, 0.4times the wing length, and a longer fin, 0.7 times the wing length.

Figure15shows plots of the side-force and yaw-moment coef-ficients against fin sweep angle for two fin lengths. Note thatFigs.15a,15c, and15dhave the same y-axis scales. The top portionof Fig. 15 shows the rigid and flexible (15,000 ft) yaw stabilityderivatives for the short fin with its length equal to 0.4 times theGoland wing length. The rigid side-force derivative (Fig. 15a)

decreases with increasing fin sweep because fin sweep reduces thelift-curve slope. [The value of the lift-curve slope of a rigid sweptsurface decreases approximately in proportion to cos().] Thisaccounts for this decrease in side-force created by sideslip. Theflexible fin side-force stability derivative decreases as well, but theplot also shows that at small sweep angles the flexible fin is moreeffective, but when the fin is swept the flexible surface is lesseffective. At largersweepangles, finbendingcoupling with finsweepdecreases the effectiveness of the fin.

The yaw-moment flexible derivative shows similar behavior to theflexible side-force derivative. However, note that the rigid stabilityderivative initially decreases (more stability) but the curve contains alocal minimum at approximately 20 deg. As the fin is swept, the finlift-curve slope decreases while the moment arm between the finaerodynamic center and the fuselage center of gravity increases.

Initially, the moment arm increase is greater than the lift-curve slopedecrease, and so the fin provides more yaw stability. However,for larger fin sweep angles, as sweep angle increases, fin bendingcoupling with fin sweep decreases the effectiveness of the fin.

The rigid surface yaw derivatives for the larger, longer fin changein a similar fashion to the shorter fin rigid yaw derivatives. However,the crossing (the point where the flexible surface becoming lesseffective than the rigid surface) occurs at a lower sweep angle. Also,at large sweep angles, 35 to 45 deg, the smaller fins flexible side-force derivative magnitude is greater (more negative) than the largerfins. Depending on the sweep angle of the fin, this may become aproblem for vehicles relying on tip-fins for directional stability.

Figure 16 plots thetwo rigid yawderivatives against both finsweepangle and fin size showing the effects of fin sweep angle on thederivative values and illustrating the trade-off between moment armlength and lift-curve slope decrease. This figure indicates that forincreased side-force the sweep angle of the fin should be minimized,while for increased yaw-moment stability the fin should be sweptback slightly to increase the moment arm of the restoring momentprovided by the vertical stabilizer.

Figure17shows the rigid stability derivatives when both the wingand the tip-fin are swept together, the sweep angle of each surface isthe same in every case.

Fig. 14 Comparison of the directional rigid and flexible stability derivatives.

Fig. 13 Aeroelastic effects cause wing lift re-distribution on a flexible

35 deg sweptback wing.



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Figure17shows that sweeping both surfaces further exacerbatesthe trade-off between the side-force and yaw-moment derivatives.The side-force derivative is maximized when the sweep angle is0 deg; once again, this is due to the rotation of the wing whichreduces the lift-curve slope. On the other hand, sweeping the wingbackwards increases the yaw-moment arm and so increases the yaw-moment stability derivative. The magnitudes of the rigid side-force

derivatives, when comparing the fin-sweep-only case (Fig.16) tothewing-and-fin-sweep case (Fig.17), are the same. Sweeping the winghas no effect on the side-force derivative as it is only a function of finsize andsweepangle. On theotherhand,the yaw-moment derivatives

increase by a factor proportional to the increase in moment armlength due to sweeping the wing. Note that this larger moment armcancels the lift-curve slope decrease effect seen when only the fin isswept.

Figure18summarizes both the results of the studies for flutterspeed and the lateral stability derivative changes with increasing finlength with a 90 cant angle. This figure shows that, for the unswept

wing and unswept fin, as the fin length increases the yaw stabilityincreases (in the ASTROS coordinate system the derivative becomesmore negative as the system becomes more stable). Note that the yawstabilitycurveis that fora rigid finand that flexibility will modify this

Fig. 15 Yaw stability derivatives, two fin lengths, a), c) side-force derivative, b), d) yaw-moment derivative.

Fig. 16 Surface plotsrigid yaw stability derivatives, increasing fin length and sweep angle.



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curve, depending on the flight altitude. For the same range of tip-finlengths, the flutter speedinitially decreases, encountersa flutter mode

switchthat increasesthe flutter speed,and then finally declines again.These competing and conflicting interests need to be reconciled if avehicle with tip-fins is developed.

V. Conclusions

Placing fins on wing tips as a substitute for a vertical tail createstwo aeroelastic problems, one dynamic, the other static. The resultspresented in this paper show that the tip-fin size, primarily the tip-finlength, and its cant angle are important design parameters that affectflutter speed. The reader is reminded that the boundary conditionsused for these studies preclude the effects of body freedom orantisymmetric modes of flutter.Smallfinstend to reduce flutter speedwhile mode switching may allow very long fins to increase flutterspeed. Static aeroelastic effects also change vehicle directional/yaw

stability, primarily through changes in lifting surface effectiveness, awell-known phenomenon for swept and unswept control surfaces.These effects are also driven by tip-fin size compared to thewing sizeand by wing sweep.

Acknowledgments

The authors gratefully acknowledge P. C. Chen and the ZONACorporation for the use of the ASTROS code. In addition, NedLindsley and Ed Pendleton of AFRL, Air Vehicles Directorategraciously supplied helpful comments and guidance throughout thisresearch.

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