Author's Accepted Manuscript

Flutter Analysis of an Articulated High Aspect RatioWing in Subsonic Airflow

A.V. Balakrishnan, Amjad Tuffaha, Iylene Patino,Oleg Melnikov

PII: S0016-0032(14)00111-2DOI: http://dx.doi.org/10.1016/j.jfranklin.2014.04.010Reference: FI2024

To appear in: Journal of the Franklin Institute

Received date: 9 November 2011Revised date: 12 March 2014Accepted date: 13 April 2014

Cite this article as: A.V. Balakrishnan, Amjad Tuffaha, Iylene Patino, Oleg Melnikov,Flutter Analysis of an Articulated High Aspect Ratio Wing in Subsonic Airflow, Journalof the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2014.04.010

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Flutter Analysis of an Articulated High Aspect RatioWing in Subsonic Airow

A.V. Balakrishnan, Amjad Tuffaha, Iylene Patino and Oleg Melnikov

May 9, 2014

Abstract

We present a methodology for calculating utter speeds of a high aspect ratio ying wing articulatedwith point masses in inviscid air ow. This highly exible wing conguration typically models a HALE(High Altitude Long Endurance) UAV (Unmanned Aerial Vehicle) type aircraft. To demonstrate the pro-cedure, we perform utter analysis on an actual articulated wing model and we investigate the dependenceof the utter speed on the number of loads mounted onto the structure and the number of panels compris-ing the ying wing for both varying and constant span. The results show that the utter speed decreasesas more panels and point masses are incorporated into the ying wing. On the other hand, the number ofpoint masses mounted onto the structure has a small effect on the utter speed if the wing span is keptconstant.

1 Nomenclaturel = wing span f tb= length of half chord f tm= wing density lb/ f ts= position along the span of the wing f tsi = ith node along the span of the structurex = chord wise positiont = time sech(s, t) = plunge variable f t(s, t) = pitch variable radI = moment of inertia lb. f tS = coupling parameterEI = bending stiffness lb. f t2

GJ = torsional stiffness lb. f t2

a= location of the elastic axis relative to the chord = air density lb/ f t3U = free stream velocity f t/secM = Mach number

This work was supported by nsf grant no. ECCS0722750A.V. Balakrishnan, Prof., Dept. of Elec. Eng., University of California, Los Angeles CA, bal@ee.ucla.eduA. Tuffaha, Ast. Prof., Dept. of Mathematics, The Petroleum Institute, Abu Dhabi, UAE, atuffaha@pi.ac.aeIylene Patino, Dept. of Elec. Eng., University of California, Los Angeles, CA, impatino@ucla.eduOleg Melnikov, Dept. of Mathematics, University of California, Irvine, CA, melnikov@ucirvine.edu

L(s, t) = liftM(s, t) = moment= Laplace transform variableli = the distance from pitching axis to the mass mi along the chordri = the radius of gyration for mass mi

2 IntroductionDesigning dependable high altitude long endurance aircrafts known as (HALE) has become essential forreconnaissance and surveillance operations. This in turn has sparked an interest in reliable light slenderwing designs, which can sustain heavy pay loads and many days and even months of non-stop ying at veryhigh altitudes. The implication of using light slender material is serious when it comes to stability and utterconsiderations. Such aircrafts enjoy high lift-to-drag ratios and can undergo serious deformations while inight, which makes them vulnerable to failure at high altitudes. We consider a special type of designwhich is the Helios UAV prototype developed by NASA under the Environmental Research Aircraft andSensor Technology program. Since the mishap of 2004 involving the HP03 model, there has been a seriesof studies concerning aeroelastic stability of ying wing congurations, following recommendations by theNasa technical report into the mishap [15]. The report asserted that Lack of adequate analysis methods ledto an inaccurate risk assessment of the effects of conguration changes leading to an inappropriate decisionto y an aircraft conguration highly sensitive to disturbances was a root cause of the mishap.

The Helios model is a HALE aircraft falling under the category of ying wing congurations, andconsists of several joined panels made of composite materials. The aircraft is powered by several electricdirect current throttle engines mounted under wing and uses solar energy during the day and a hydrogen-airsystem during the night. The aircraft is controlled remotely from the ground by a pilot and is designed toy at up to 100,000 ft altitude at low speeds in the range of 20-40 ft/sec. The HP03 model in particularrepresents the fth generation of HALE ying wing aircrafts designed by NASA, and has a wing span of247 ft comprising six panels (each about 41 ft long) and a chord length of 8 ft with 11.5 inches thickness,while the exibility of the wing allows for the formation of a dihedral U shape during ight. Accordingto the Nasa technical report on the HP03 mishap, the aircraft experienced an increased dihedral angle inturbulence and then underwent rapid oscillations which were possibly exacerbated by gust, resulting instructural failure followed by a crash into the pacic ocean.

Many works in the literature have treated highly exible wing designs and their ight dynamics in detailand have analyzed the stability, both dynamic and static, of various types of highly exible wing structures.A detailed treatment of the modeling aspects as well as stability studies for different types of HALE aircraftincluding roll and gust response can be found in [3]. In [16], the author conducts detailed analysis of theaeroelastic response of a typical high aspect ratio wing representative of a HALE aircraft along with acomparison with experimental wind tunnel data. The analysis relies on nite element analysis of nonlinearbeam models and ONERA codes to account for the aerodynamics. In [5, 6], Dowell and Tang carry outa thorough theoretical and experimental study of LCO and the gust response of a high aspect ratio wingrepresentative of a HALE aircraft.

Flying wing congurations, which is our main focus, were considered by Patil and Hodges who haveanalyzed rigid body motion instabilities and conducted a trim analysis of a particular highly exible yingwing conguration representative of the Helios model [4]. Su and Cesnik have also studied ight dynamicstability and response of a similar ying wing conguration in [2], using a nonlinear structure model toconsider body freedom utter and the gust response at different altitudes.

In this paper, we develop a methodology for calculating utter speeds for an articulated wing structurecomprising several elastic beams. The ying wing model under consideration corresponds to the Helios

prototype HP03 with the same conguration as in [4, 2], but our interest is mainly utter and aeroelasticstability. In contrast to all these works in the literature which rely on available CFD codes to determinethe aerodynamic loads and FEM to solve for the structural dynamics, our approach is the more recentlydeveloped continuum model approach [1]. In a recent paper [12], the authors conduct a comparison ofresults obtained from the continuum model [1] with results obtained using the NATASHA software, for thebasic Goland beam model [9], and in particular report an agreement in utter frequencies of the rst fourmodes. We intend to extend the continuum model approach [11] to address the case of the articulated yingwing case and the effect of engine placement.

Our analysis relies on the fact that these slender ying wings have very low natural frequencies and yat relatively low speeds [2], so that it is reasonable to assume a Mach number M = 0. The structural andphysical parameters we use are also the same as in [4, 13] and for simplicity we only consider uniformlydistributed pods (propulsive units) represented by point masses in the model along the pitch axis of thestructure. The parameters used are indeed reective of the Helios ying wing congurations, and forinstance the HP03 model which experienced the failure is comprised of 247 feet long wing with a chordlength of about 8 feet with 6 propulsive units, while earlier generations were lighter and had shorter wingspans [15]. The model we use is a Goland elastic beam model with two degrees of freedom, plunge andpitch, which is appropriate for high-aspect ratio wings. The model comprises a sequence of joined beamsand allows for exible movement at the joints and for a dihedral angle formation at the ends. The presenceof the pods at the joints of the connected beams necessitates modeling the structure dynamics using thearticulated beam model following Goland [10, 8, 9, 7]. As for aerodynamics, we rely on an analyticalsolution of the linearized Possio equation in inviscid airow for Mach number M = 0 which yields thelift and moment forces as functions of the plunge and the pitch. Important developments in the area ofmathematical aeroelasticity in the past few years have made it possible to consider such a continuum modelapproach to utter analysis as an alternative to the CFD approach [11].

The methodology for calculating utter speed relies on the usual tracing of the root locus of a relevantstructure aeroelastic mode with the varying free stream velocity. However, the continuum model approachinvolves rst solving the Possio equation for the aerodynamic forces in terms of the structure state variable,in order to determine the structural dynamics, without discretizing any of the equations in contrast to thedominant approach. We then proceed to identify the relevant aeroelastic structure modes, and trace theirstability with the varying speed parameter. In this context, we provide precise mathematical denitions ofthe aeroelastic modes and the utter speed following earlier works in the literature [11].

The methodology was then implemented in a Matlab program which calculates the stability curve forthe relevant structure modes given physical parameters and wing specications. Computationally, the aeroe-lastic modes are roots of a determinant function involving a number of exponential matrices matching thenumber of panels comprising the wing, and the simple Matlab program traces an aeroelastic mode with thechanging speed parameter until instability occurs. As an example, we provide the results of applying thismethodology to perform utter and aeroelastic stability analysis for the particular articulated wing struc-ture studied by Patil and Hodges [4]. We also use the results to show the dependence of utter speed onthe number of panels (beams) comprising the structure, the wing span, and the number of loads (engines)mounted onto the structure.

We nally note that the prospect of a successful control design of subsonic wing utter in the futurewill closely depend on the development of a sound theoretical framework for the analysis and prediction ofutter, which would enhance the current CFD approach.

3 The Mathematical Model of the Flying WingWe consider a uniform beam model with two degrees of freedom, plunge/bending h(s, t) and pitch/torsionangle (s, t) for 0< s< l where l is the total wing span and t is the time variable. Point masses denoted miare at discrete points (nodes) along the span s = si, i = 0,1, ...,n,n+1 with s0 = 0 and sn+1 = l (see gure1). The beam equations are

S(s, t)+mh(s, t)+EIh(s, t) = L(s, t), (1)

Sh(s, t)+ I (s, t)GJ (s, t) =M(s, t), (2)

where si1 < s < si and S is the coupling parameter.Here, the forcing terms L(s, t) and M(s, t) in (1) and (2) are the lift and moment forces acting on the structurewhich we discuss in the next section on Aerodynamics. The superdots denote time derivatives while primesdenote spatial derivatives in the same notation as in [10]. Moreover, the structure parameters EI, GJ andI denote the bending rigidity, the torsional rigidity and the moment of inertia respectively while m denotesthe mass density of the wing along the span (mass per unit length) (see table 1 below).

We allow for discontinuities at the points si in (s, t) and h(s, t). In particular, supplementing (2) atthe nodes si, we have the conditions:

mir2i (si, t)+milih(si, t)GJ( (si+, t) (si, t)

)+ui(t) = 0, (3)

where the ui() are controls, if any, at the nodes si for i= 0, ...,n+1, while ri is the radius of gyration and liis the distance (normal) from pitch axis to the point mass mi along the chord at s = si. We refer the readerto [7] for detailed derivations of these boundary conditions.

On the other hand, supplementing (1) at the nodes s= si, we have the condition

mih(si, t)+mili(si, t)+EI(h(si+, t)h(si, t)

)+ui(t) = 0 (4)

The boundary conditions are the free-free end conditions

(0, t) = 0 (5) (l+, t) = 0 (6)

and

h(0, t) = 0 h(0, t) = 0 (7)h(l, t) = 0 h(l+, t) = 0. (8)

The ying wing under consideration here has a span of 40400 ft long with pods or uniformly spacedpropulsive units 40 ft apart along the wing as in the gure shown.

Each propulsive unit or point mass weighs 60 lb. The structural parameters used are listed in table 1above following [4]. Moreover, we only consider the symmetric case where li = 0 and ri = 0 which meansthat the pods are placed along the pitching axis.

4 The AerodynamicsWe need only consider the linearized typical-section (airfoil) aerodynamics and hence we can follow thedevelopment in [11] closely.

Figure 1: Diagram of the Wing Baseline Geometry

Table 1: Structure Parameters

2b Chord length 8 f tGJ Torsional rigidity 0.4106 lb. f t2EI Bending rigidity 2.5106 lb. f t2

Bending rigidity (chordwise) 30106 lb. f t2m Mass per unit length 6 lb/ f t

Pitch axis location a=0.25, 25% chordI Centroidal mass moment of inertia about x axis (torsional) 30lb. f t

About y axis 5 lb. f tAbout z axis 25 lb. f t Air density 0.0023769 slugs/ f t3

Thus, we begin with the downwash function (normal velocity of structure) which is given by:

wa(x,s, t) =(h(s, t)+(xab)(s, t)+U(s, t)) (9)

where ab is the location of the elastic axis, b is the half-chord andU is the far-eld air speed. Then for xeds (point along the span), the lift and the moment forces are given by

L(s, t) =U bb

A(x, t)dx (10)

M(s, t) =U bb(xab)A(x, t)dx (11)

respectively, where A(x, t)s Laplace transform

A(x, ) = 0

e tA(x, t)dt, Re( )> a

( > 0) is the solution of the Possio integral equation

wa(x, ) = bb

P(x , )A( , )d , |x|< b (12)A(x, ) = 0, |x|> b (13)

Physically, A(x, t) at any xed point s along the wing span corresponds to the pressure jump across the wingand is reasonably assumed to be proportional to the acceleration potential of the disturbance ow [1].

The spatial Fourier transform of the kernel P is given by

eixP(x, )dx =12

1bU + i

M2

2b2

U2+2

bU

M2i +(1M2)2 (14)

while from (9) we have

wa(x, ) = 0

e twa(x, t)dt

=( h(s, )+ x (s, )+ (s, )(U ab )) (15)

with h(s, ), (s, ) denoting the Laplace transforms of h(s, t) and (s, t) respectively, while the initialconditions are set to zero. A convenient and generally accepted normalization here is to dene k = b/U(reduced frequency) so that we may take b = 1. Therefore, the Possio equation can be reintroduced in newvariables dened on 1< x < 1 as

wa(bx, ) = 11

P(x ,k) A( , )d , |x|< 1 (16)

where the new kernel P is dened as P(x,k) = P(bx, ) and the new variable A(x, ) = bA(bx, ).We shall use an existing solution of the Possio equation for M = 0 as function of the plunge and pitch

to feed into the structural equations, following [11].In order that we obtain the actual solution A given the downwash wa dened in (15), we take advantage

of the nature of the dependence of wa(bx, ) on the variable x and the linearity of the problem. Observethat it is enough that we solve the Possio integral equation (16) for A when wa = fi and i= 1,2 where

f1(x) = 1, |x|< 1f2(x) = x, |x|< 1.

We then let A1 and A2 denote the solutions of the Possio equation (16) corresponding to left hand side of f1and f2 respectively, for every > 0. Therefore, we have

A( , ) =( h(s, )A1( , )+bA2( , ) (s, )+ A1( , )(s, )(U ab )) (17)Dening the wi j functions by

w11(M, ) = 11

A1( , )d (18)

w12(M, ) = 11

A2( , )d (19)

w21(M, ) = 11

A1( , )d (20)

w22(M, ) = 11

A2( , )d . (21)

then substituting (17) into (10) and (11), the lift and moment forces in Laplace domain can be expressed interms of the wi j functions as

L(s, ) =bU2(

kbw11h(s, )+(kw12+(1ak)w11)(s, )

)(22)

M(s, ) =U2b2(

kb(w21aw11)h(s, )+(kw22+(1ak)w21akw12a(1ak)w11)(s, )

).

(23)

The wi j can be calculated explicitly for M = 0 using solutions to the Possio equation (16) and are given in[11] as explicit functions. Unfortunately, closed form solutions of the Possio equation for 0 < M < 1 arestill an open problem [1], but for the low speed conguration that we are considering, it is reasonable to usethe M = 0 solutions.

5 Aeroelastic Modes and StabilityWe are now ready to consider the problem of stability of the aeroelastic system. We begin by taking Laplacetransforms in (1) and (2), setting all initial conditions to zero. Thus, we have

2S(s, )+ 2mh(s, )+EIh(s, ) = L(s, ), si < s < si+1 (24)

2Sh(s, )+ 2I (s,)GJ (s, ) = M(s, ), si < s < si+1. (25)

Laplace transforming (3), we have

2(r2i mi(si, )+ limih(si, )

)GJ( (si+, ) (si, ))= 0 (26)or,

GJ (si+, ) = GJ (si, )+ 2(r2i mi(si, )+ limih(si, )

). (27)

Similarly, Laplace transforming (4) yields

EIh(si+, ) = EIh(si, ) 2mi(h(si, )+ li(si, )

)(28)

while the boundary conditions (5)(8) become

(0, ) = 0 (29) (l+, ) = 0 (30)

and

h(0, ) = 0 h(0, ) = 0 (31)h(l, ) = 0 h(l+, ) = 0. (32)

We should note that due to these boundary conditions, the conditions (27) and (28) at s= 0 become

GJ (0+, ) = 2(r20m0(0, )+ l0m0h(0, )

)(33)

EIh(0+, ) = 2m0(h(0, )+ l0(0, )

), (34)

and at s= l,

0= GJ (l+, ) = GJ (l, )+ 2(r2n+1(l, )+ ln+1h(l, ))mn+1 (35)0= EIh(l+, ) = EIh(l, ) 2mn+1(h(l, )+ ln+1(l, )). (36)

Fixing , we have here a two-point boundary value problem for a system of ordinary differential equa-tions with s as the independent variable. We may now invoke state space theory. Thus, let

Y (s, ) =

h(s, )h(s, )h(s, )h(s, )(s, ) (s, )

,

for 0 < s < l. After substituting the expression for L and M in (22) and (23) given in terms of the statevariables h and , the system of equations (24)-(25) can be expressed using the state equation

Y (s, ) = A( )Y (s, ), si < s < si+1

where A( ) is given by

A( ) =

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0w1 0 0 0 w2 00 0 0 0 0 1w3 0 0 0 w4 0

and

w1 = 1EI( 2m+Ubw11

)(37)

w2 = 1EI( 2S+bU2((1ak)w11+ kw12)

)(38)

w3 =1GJ

( 2S+b2U (w21aw11)

)(39)

w4 =1GJ

( 2I +b2U2(w21+ kw22a(1ak)w11ak(w21+w12))

). (40)

Let Q be the 63 matrix

Q=

1 0 00 1 00 0 00 0 00 0 10 0 0

,

and let

Ei =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

2miEI 0 0 1 2limiEI 0

0 0 0 0 1 0 2limiGJ 0 0 0

2r2i miGJ 1

, i= 0,1, ...,n+1.

where Ei is the transition matrix across a given node si, relating the state at si+ to that at si as determinedby equations (27) and (28). In other words, Y (si+) = EiY (si).

Then, at s1 we have

Y (s1, ) = es1A( )E0Q h(0, )h(0, )

(0, )

due to the boundary conditions (29) and (31) and the transition equations (33) and (34).In general, the state Y (s) across the si nodes can be expressed as

Y (si, ) = eA( )(sisi1)Y (si1+, )Y (si+, ) = EiY (si, ), i= 1, ...,n

Y (l, ) = eA( )(lsn)Y (sn+, )Y (l, ) = En+1Y (l, ).

We now denote by P the 36 matrix:

P=

0 0 1 0 0 00 0 0 1 0 0

0 0 0 0 0 1

.

Hence, by the boundary conditions (30) and (32) prescribed at the end s= l, we must have

PEn+1eA( )(lsn)EneA( )snsn1 ....E1eA( )s1E0Q

h(0, )h(0, )

(0, )

=

h(l, )h(l, )

(l, )

=

00

0

. (41)

If we let

d(M, ,U) = det(PEn+1eA( )(lsn)EneA( )snsn1 ....E1eA( )s1E0Q

), (42)

then (41) implies that

d(M, ,U) = 0, (43)

which generalizes a similar result in [10], without point masses. Note that if mi = 0 we have Ei = I,i = 0,1, ...,n,n+ 1. For xed M, the roots of (43) are the aeroelastic modes, and the correspondingsolution Y (, ) yields the mode shapes.

5.1 Structure ModesThe structure modes are obtained by settingU = 0 in (43), and solving for the roots of d(M, ,0) = 0. Notethat M plays no role here, since we are assuming a xed M = 0 which is again appropriate for low speedying wings.Free-Free Structure Modes:

If we specialize to the case of no controls and no point masses, so that mi = 0 and set U = 0 and S = 0,we obtain

d(M, ,0) =12

4(1+ cosh l cos l)sinh l = 0 (44)

where

= (w1)1/4

=

w4

w1 = 2mEI

w4 = 2IGJ

.

The determinant factorizes into two factors and we distinguish the zeros of each factor. The zeros of1+ coshl cosl = 0 are the bending modes and the zeros of sinh l = 0 are the torsion modes.

Setting = i , the bending mode frequencies are given by

n =(xn

l

)2EIm

where xn are the roots of1+ cosxn coshxn = 0

and are also ordered by increasing magnitude. On the other hand, the torsion modes are

wn =nl

GJI

, n= 1,2, ...

The corresponding mode shapes are obtained by solving (43) and obtaining the corresponding Y (, ):

Y (s, ) = eA( )sQ

h(0, )h(0, )

(0, )

.

We should note that = i = 0 is a root of (43) but the corresponding mode shape has zero elastic energyand is classied as a rigid-body mode.

6 Root Locus/Flutter SpeedTo determine aeroelastic stability, we start with the structure modes which were determined above by settingU = 0 in (43). For U > 0, which is our main concern, the roots of (43) (aeroelastic modes) continue to becountable and can be located by tracing the root locus using a computer program, taking advantage of thefact that this determinant function for each U is analytic in (omitting the line 0). Moreover, for each , d(M, ,U) is analytic in U and we may invoke the implicit function theorem, which implies that theroots are analytic functions of U , n(U). Thus, if n(0) denote the zeros of d(M = 0, ,U = 0) which arethe same structure modes for S = 0, then

Re(n(0)) = 0; n(0) = in

and since the modes are not coupled we may use the usual terminology of bending modes and torsionmodes. For S = 0, the modes are coupled but for small S which is characteristic of the beams to whichwe shall limit ourselves, the coupling is small enough so that we may still talk in terms of (predominately)bending modes and torsion modes [4]. Note that these modes are ordered by increasing magnitude.

The utter speed UF is then dened as the minimum speed at which any of the modes n exhibit utter[11]. In other words UF = inf{U : Re(n(U)) = 0}.

7 Flutter Analysis and ResultsWe now use the above methodology to carry out utter analysis for the ying wing conguration of gure 1with the parameters listed in table 1. In particular, we trace the root locus for the rst bending, second bend-ing modes as well as the rst and the second torsion modes with the varying free stream speed parameter

Figure 2: Flutter Analysis Flow Chart

U . This again entails nding the root of the determinant in (43) using a root nding algorithm which usesthe particular structure mode as an initial value and marching with the speed parameterU , until the real partof the root changes sign from negative to positive. Using a simple program in Matlab, this strategy wasimplemented to nd the root locus for the relevant modes. The program computations are completed in fewseconds on a 2.8 GHz Intel(R) Core (TM)i7 2640M processor with 4 GB of RAM, and the program allowsfor any choice of parameters, number of panels (beams) or number and mass of equally spaced loads at thejunctions between the beams.

In the rst part of the study, we implemented the program with the parameters in table 1, to examinethe variation of the utter speed for the rst and second bending and torsion modes with the number ofpanels comprising the wing (all else constant). In particular, we increase the span each time by adding anadditional 40 ft long beam, along with a 60 lb point mass at the junction, starting with a single 40 ft Golandbeam with no loads.

While we considered the simple case of equally spaced loads and similar beams, the program can beeasily modied to address variation in panel sizes and unequal spacing. The justication of this study ismotivated by the different Helios models with later models comprising additional panels and loads.

The result of the implementation shows that the rst and the second bending modes exhibit utter ata critical speed which decreases each time an additional 40 ft panel and an additional 60 lb point massmounted at the panel junction are added. Figures 3 and 5 show this relationship with the number of 40 ftpanels comprising the wing plotted on the horizontal axis and the utter speed on the vertical axis. Thenumber of loads (point masses) in each case is equal to the number of panels +1, since a load is mounted

on each junction.The utter speed goes from 134 ft/s to as low as 10 ft/sec as the above described variation is imple-

mented. However, table 2 and gure 5 contain an outlier, which is the bending mode for a 120 ft wingconsisting of three 40 ft panels articulated with two 60 lb loads, where there was no evidence of utter. Theresult in this particular case in fact coincides with the rst torsion mode, as it happens that the second bend-ing structure mode and the rst torsion mode are quite close at this conguration as seen in gure 15, whichshows all the natural frequencies of the structure with the varying number of (panels) beams comprising thestructure.

An example of the root locus which captures the real part of the Aeroelastic bending modes, versusthe free stream velocity is shown in gures 7 and 8 for the case of 240 ft ying wing with 5 point massesuniformly placed along the wing. The utter speeds are 22.83 and 34.54 ft/s. This case is of particularinterest since it reects approximately the Helios HP03 model which underwent the mishap in 2003 [15].

The bending mode frequencies in Hertz as well as the normalized frequencies k = b/U correspond-ing to the utter point are also plotted in gures 4 and 6 as functions of the wing span along with thefrequency and show a decreasing pattern. As expected, these frequencies are low falling in the range of0.10.35.

For structures comprising three 40 ft beams or less, the torsion modes do not exhibit utter in the speedrange examined, but torsion utter occurs at relatively higher speeds for wing spans above 160 ft. Tables 4and 5 show utter speeds and frequencies for 1st and 2nd torsion modes. Figure 12 shows the root locus ofan 80 ft wing consisting of 2 beams and one point mass for the rst torsion mode while gure 14 shows theroot locus for a 240 ft wing consisting of 6 beams and 5 point masses.

In the second part of the study, we consider a wing of xed span of 240 ft with a varying number ofbeams and loads (60 lb point mass) at the junctions, starting with the basic 240 ft beam with no loads, thentwo 120 ft beams with one load in the middle, and so on. This variation in the number of loads and beamsin the 240 ft wing causes the utter speed to decrease slightly, for the particular symmetric case consideredhere as seen in gures 9 and 10. The effect becomes noticeable with about 5% change after incorporating10 additional loads as seen in gures 9 and 10. Figures 11 and 12 depict how the utter frequencies changeas the number of panels and loads comprising the 240 ft wing in the case of the rst and second bendingmodes. The gures also shows the natural frequencies for the 240 ft wing and how they change with thenumber of beams and loads comprising the wing. Tables 2 and 3 list utter speeds and utter frequenciesfor a ying wing with varying number of panels and point mass loads.

8 ConclusionAlgorithms based on continuum aeroelastic models are found to be effective in performing utter analysisand can enhance the current CFD approach. The particular methodology presented for computing utterspeeds and performing stability utter analysis for an articulated ying wing in inviscid air ow [11] wasapplied to a specic articulated structure model appropriate in describing HALE UAV type aircrafts withhighly exible wings and has given reliable results.

The algorithm depends on a continuum model formulation of both structure and aerodynamics and wasimplemented using a Matlab Program which computes the root locus for any given mode of the structure.

We implemented the program to examine the variation of utter speed and utter frequency with thenumber of panels and loads mounted onto the wing, for parameters and congurations capturing the He-lios aircraft prototypes. The results conrm that utter is exhibited at low speed range (under 40 ft/sec)

Table 2: Flutter Speed and Frequency (1B)

Number of 40 ft Panels Structure 1st Bending Mode Flutter Speed (ft/s) Normalized Flutter Frequency (k)1 ( No point masses) 9.026 134.0 0.2923

2 2.128 69.54 0.14703 0.9285 46.10 0.12074 0.5177 34.41 0.10705 0.3296 27.45 0.09946 0.2281 22.83 0.09497 0.1672 19.55 0.09188 0.1278 17.09 0.08979 0.1008 15.19 0.088210 0.0816 13.67 0.08711 0.0673 12.44 0.086112 0.0565 11.42 0.085613 0.0481 10.55 0.0858

Table 3: Flutter Speed and Frequency (2B)

Number of 40 ft Panels Structure 2nd Bending Mode Flutter Speed (ft/s) Normalized Flutter Frequency (k)1 ( No point masses) 24.88 112.7 0.8888

2 5.865 103.1 0.24993 2.560 4 1.427 52.22 0.13625 0.9085 41.59 0.12116 0.6287 34.54 0.11127 0.4608 29.54 0.10458 0.3522 25.8 0.09889 0.2778 22.92 0.096310 0.2248 20.00 0.093711 0.1856 18.75 0.091712 0.1558 17.20 0.090213 0.1327 15.90 0.0893

Table 4: Flutter Speed and Frequency (1T)

Number of 40 ft Panels Structure 1st Torsion Mode Flutter Speed (ft/s) Normalized Flutter Frequency (k)5 1.814 55.98 0.14687 1.296 39.69 0.12158 1.134 43.59 0.130010 0.9069 58.42 0.148511 0.8245 38.02 0.122212 0.7557 34.82 0.116813 0.6976 32.1 0.1125

Table 5: Flutter Speed and Frequency (2T)

Number of 40 ft Panels Structure 2nd Torsion Mode Flutter Speed (ft/s) Normalized Flutter Frequency (k)4 4.535 86.48 0.23257 2.591 69.77 0.17748 2.267 61.74 0.163710 1.814 56.72 0.1554

for wings more than 200 ft long with several loads (The Nasa technical report [15] mentions that HeliosHP03-2 aircraft which experienced the failure was ying at a speed of 37 ft/sec just before it experiencedthe malfunction). The next necessary step in the analysis should involve incorporating the dihedral angleformation into the model and the algorithm in order to further enhance the aeroelastic analysis of Heliostype aircrafts. Moreover, ner root nding algorithms of complex valued functions can further improvethe program and better separate the modes especially for higher aspect ratio wings where the bending andtorsion modes can become more difcult to separate. To further support the continuum approach and givecredibility to the results, a comparison with the CFD data for the same case study would be helpful, as wasdone with the single Goland beam case in [13].

References[1] BALAKRISHNAN, A.V. ; Aeroelasticity: The Continuum Theory, Springer, New York, 2012.

[2] SU, W. & CESNIK, C. E. S.; Dynamic Response of Highly Flexible Flying Wings, 47thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 1 - 4 May2006, Newport, Rhode Island.

[3] SU, W.; Coupled Nonlinear Aeroelasticity and Flight Dynamics of Fully Flexible Aircraft; Ph.DThesis, University of Michigan, 2008.

[4] PATIL, M. & HODGES, D.; Flight Dynamics of Highly Flexible Flying Wings, In Proceedings of theInternational Forum on Aeroelasticity and Structural Dynamics, Munich, Germany, June 2005.

[5] TANG, W. D. & DOWELL, E.; Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings, AIAA JOURNAL, VOL. 39, NO. 8, 2001, PP. 14301441.

[6] TANG, W. D. & DOWELL, E.; Experimental and Theoretical Study of Gust Response for High-Aspect-Ratio Wings, AIAA JOURNAL VOL. 40, NO. 3, MARCH 2002.

[7] RUNYAN, R.L. & WATKINS, C.E.; Flutter of a uniform wing with an arbitrarily placed mass ac-cording to a differential equation analysis and a comparison with experiment, NASA TECHNICALREPORT, NACA TN 1848, 1949.

[8] GOLAND, M. & LUKE, Y.L.; A Study of the Bending Torsion Aeroelastic Modes for Airplane Wings,J. OF AERONAUTICAL SCIENCES, VOLUME 16, NO. 7, JULY 1949, PP. 389396.

[9] GOLAND, M. & LUKE, Y.L.; The Flutter of a Uniform Wing with Tip Weights, J. OF APPL. MECH.,VOLUME 15, NO. 1, MARCH 1948, PP. 1320.

[10] BALAKRISHNAN, A.V.; Dynamics and Control of Articulated Anisotropic Timoshenko Beams. INH. S. TZOU AND L.A. BERGMAN, EDS, DYNAMICS AND CONTROL OF DISTRIBUTED SYSTEMS.CAMBRIDGE UNIVERSITY PRESS, 1998, PP. 121-201.

[11] BALAKRISHNAN, A. V. & ILIF, K. W.; A Continuum Aeroelastic Model for Inviscid Subsonic WingFlutter, JOURNAL OF AEROSPACE ENGINEERING, JUNE 2007.

[12] MARDANPOUR, P., HODGES, D.H., NEUHART, R. & GRAYBEAL, N.; Effect of Engine Placementon Aeroelastic Trim and Stability of Flying Wing Aircraft, AIAA, 2012.

[13] SOTOUDEH, Z., HODGES, D.H. & CHANG, C.S.; Validation Studies for Aeroelastic Trim and Sta-bility Analysis of Highly Flexible Aircraft. JOURNAL OF AIRCRAFT 47(4), P. 12401247, 2010.

[14] PATIL, M.J., HODGES, D. & CESNIK, C. E. S.; Nonlinear aeroelastic analysis of complete aircraftin subsonic ow. JOURNAL OF AIRCRAFT 37(5):753 NO. 760, SEPOCT 2000.

[15] NOLL, T. E., BROWN, J. M., PEREZ-DAVIS, M. E., ISHMAEL, S. D., TIFFANY, G. C. & GAIER,M.; Investigation of the Helios Prototype Aircraft Mishap, VOLUME 1, MISHAP REPORT, NASAREPORT, JAN. 2004.

[16] JAWORSKI, J.; Nonlinear Aeroelastic Analysis of Flexible High Aspect Ratio Wings Including Corre-lation With Experiment, PH.D. THESIS, DUKE UNIVERSITY, 2009.

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