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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, [email protected]. Tuffaha, Ast. Prof., Dept. of
Mathematics, The Petroleum Institute, Abu Dhabi, UAE,
[email protected] Patino, Dept. of Elec. Eng., University of
California, Los Angeles, CA, [email protected] Melnikov, Dept.
of Mathematics, University of California, Irvine, CA,
[email protected]
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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
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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.
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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.
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Figure 1: Diagram of the Wing Baseline Geometry
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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.
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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, )
,
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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, )
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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)
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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
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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].
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Figure 4
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Figure 5
Figure 6
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Figure 8
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Figure 9
Figure 10
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Figure 11