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22 June 2021
POLITECNICO DI TORINORepository ISTITUZIONALE
A comprehensive analysis of wing rock dynamics for slender delta
wing configurations / Guglieri G.. - In: NONLINEARDYNAMICS. - ISSN
0924-090X. - 69:4(2012), pp. 1559-1575.
Original
A comprehensive analysis of wing rock dynamics for slender delta
wing configurations
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PublishedDOI:10.1007/s11071-012-0369-3
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A comprehensive analysis of wing rock dynamics for slenderdelta
wing configurations
Giorgio Guglieri
Abstract The paper deals with the study of an analyti-cal model
of wing rock, based on parameter identifica-tion of experimental
data. The experiments were per-formed in the Aeronautical
Laboratory of Politecnicodi Torino, in the D3M Low Speed Wind
Tunnel, on a80◦ delta wing with a modular fuselage, designed witha
cylindrical forebody and a conical nose tip. Free-to-roll tests
have been used to determine build up andlimit cycle characteristics
of wing rock. An analyticalnonlinear model was derived. Parameters
were iden-tified by means of least squares approximation of
ex-perimental data with coherent initial conditions. Theconsistency
of time histories, reproduced by numer-ical integration, was also
analyzed. This formulationcorrectly predicts stable limit cycles
for a wide rangeof airspeeds, angles of attack, and release roll
angles.Finally, the impact of aircraft configuration on wingrock
parameters is here outlined.
Keywords Aircraft dynamics · Delta wingaerodynamics · Vortex
dynamics
Nomenclatureai nondimensional coefficientsb wing span
c wing root chordCl rolling moment coefficient ( L/qSb)Claer
rolling moment coefficient (aerodynamic term)Clf rolling moment
coefficient (friction term)f oscillation frequencyIxx model
inertiak reduced oscillation frequency ( πf b/V )� oscillation
cycleL rolling momentq dynamic pressure (ρV 2/2)S model wing
surfaceSwt wind tunnel cross sectionRe Reynolds number (based on
c)t timet̂ nondimensional time (t/t∗)t∗ reference time (b/2V )TPI
Politecnico di TorinoV airspeedWR Wing Rockα angle of attackβ angle
of sideslipμf rolling moment coefficient (friction rate
coefficient)ϕ roll angleϕ0 release roll angleΔϕ oscillation
amplitude in rollΛ sweep angleρ air density. time derivative
1
mailto:[email protected]
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1 Introduction
Wing rock is a self-sustained large-amplitude oscilla-tion in
roll that may exhibit a dynamically stable limitcycle. The final
state is generally stable and charac-terized by both large roll
attitudes and coupling withdirectional modes. This phenomenon can
seriouslylimit the operational effectiveness of aircraft
utilizinghighly swept wings during take-off, landing, and
ma-neuvering flight.
The motion has been observed in flight, but hasbeen difficult to
explain because of its similarity to alightly damped Dutch-roll
mode. The evidence sug-gests that the wing rock motion is a limit
cycle oscilla-tion wherein the amplitude and period of the motion
issolely a result of aerodynamic non-linearities. This isa contrast
to the response of a lightly damped Dutch-roll mode where the
amplitude is determined by theinitial conditions. The presence of
mechanical hystere-sis in stability augmentation systems can also
give riseto limit cycle motions, and this situation should notbe
confused with either the Dutch-roll or aerodynamichysteresis.
The main aerodynamic parameters of wing rockare: (i) angle of
attack, (ii) angle of sweep, (iii) leadingedge extensions, and (iv)
slender forebody. Therefore,the aircraft that are susceptible to
the wing rock phe-nomenon are those containing these parameters,
suchas aircraft with highly swept wings operating withleading edge
extensions. Such aircraft include manymodern combat aircraft such
as Panavia Tornado, Eu-rofighter Typhoon, F-16, F-18, and the
supersonic civiltransport aircraft Concorde as examples.
The onset of wing rock is related with a nonlinearvariation of
roll damping derivative with α, sideslipangle β , oscillation
frequency and amplitude [1].
Wing rock is primarily observed by means of windtunnel
free-to-roll experiments for very slender deltawings (leading edge
sweep Λ ≥ 75◦) at high angles ofattack (α ≥ 25◦). For these
experimental conditions,unstable roll damping is found for moderate
bank an-gles ϕ (i.e., moderate sideslip). Differently, dynamicroll
stability occurs for larger roll displacements ϕ(i.e., larger
sideslip). The combined effect of dihedralstatic stability (i.e.,
the restoring moment) and nonlin-ear roll damping is the basic
explanation for the pres-ence of the limit cycle.
Aircraft configurations with slender forebodies areaffected by
wing rock, due to the unsteady interac-
tion between primary forebody vortices and lifting sur-faces
(leading edge extensions, wing, and stabilizers).Therefore, the
oscillatory dynamics is substantially ir-regular in terms of
amplitude and frequency.
The forebody flow pattern (see [5] for a very com-plete
discussion of the subject) is mainly character-ized by a primary
pair of vortices (in bound secondaryvortical systems play a
marginal role) emanating fromthe apex and separating from the body
along the lee-ward side of the fore part of the fuselage.
Typically,the vortex pair (if the fuselage body is slender)
be-comes asymmetric for angles of attack exceeding themagnitude of
the apex angle, measured as the angleenclosed by the tangents to
the ogive nose shape (usu-ally above 30◦). This asymmetry is
present for sym-metric flight conditions and the direction of
preva-lent sideforce is determined by ogive surface
micro-asymmetries (roughness). The vortex cores (longitudi-nal axis
of the vortices) are displaced apart from thefuselage and, if the
flow is asymmetric, they inducean unbalanced interference with the
lifting surfacesdownstream (leading edge extensions, wing, and
em-pennages). These changes in the flow topology affectthe behavior
of the wing and are also believed to giverise to critical states
[4]. A critical state is defined asthe value of the motion variable
(e.g., the angle of at-tack or roll angle) where there is a
discontinuity in theaerodynamic coefficient or its derivative.
The forebody-induced wing rock may be sup-pressed either by
changing forebody cross-section andslenderness or by the adoption
of forebody vortex con-trol techniques (boundary layer
suction-blowing ormovable forebody strakes) [2]. Forebody strakes
areusually installed close to the ogive apex, either fixedor
deployable. The strakes induce the separation of theflow and the
formation of the vortices is fixed alongthe leading edge of these
nonlifting surfaces. Further-more, if deployed symmetrically, they
induce a sym-metric behavior of the forebody vortices, cancelingout
the destabilizing sideforce component. If deployedasymmetrically,
they enhance the asymmetry and canbe used as an auxiliary control,
for high angle of attackdirectional steering. In any case, due to
their negligi-ble lifting contribution, they do not change the lift
ofthe overall configuration, as extensively demonstratedin [3].
Indeed, the aerodynamic regime on these configu-rations is
dominated by vortical flows [5]. Evidence isgiven that, during wing
rock oscillations, the normal
2
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position in the crossflow plane of vortex cores is af-fected by
hysteresis. The roll angular velocity greatlyinfluences both the
pressure distribution on the wingsurface and the roll damping.
Furthermore, the vortexstrength varies during the wing rock
process. Free-to-roll and forced oscillation tests on slender delta
wingsindicated that wing rock build up is substantially pro-moted
by roll damping decrease at high angles of at-tack.
The systematic approach to the study of wing rockis based on
wind tunnel experimental investigation ofroll dynamics [6–13] for
highly swept delta wing mod-els (see [12, 13] for an interfacility
comparison). Theseexperiments were performed in nonuniform test
con-ditions (i.e., accuracy of the data acquisition system,model
size, equivalent dihedral, roll inertia, size ofthe test section,
geometry of the support, type of bear-ings, and levels of
friction). As a matter of fact, espe-cially Arena [8–10] conducted
a very thorough exper-imental study of the wing rock motion on a
flat platedelta wing. This study provides an interesting exam-ple
of the importance of unsteady aerodynamics onthe wing rock motion.
What makes the study unique isthe measurement of the unsteady
aerodynamics, sur-face pressures, and off-surface location of the
leading-edge vortices in combination with a numerical simu-lation
of the wing rock motion. Because of the com-pleteness of this
study, it was used as a referencefor the present experimental
program. The geometriesof different reference models and the
blockage fac-tors S/Swt are presented in Table 1. Wind tunnel
testsare performed with one degree of freedom free-to-rollrigs,
neglecting the typical couplings observed in realaircraft motion
dynamics. These simplified geometriesexhibit stable limit cycles
and correctly reproduce thedominant effect of primary wing
vortices. Differently,the analysis of complete aircraft roll
dynamics is quitedifficult, as the relevant aerodynamic
interactions be-tween forebody, lifting surfaces, and empennages
mayalter the onset mechanism of wing rock.
Diverse mathematical formulations of the differen-tial equation
governing the single degree of freedomapproximation of the roll
mode were suggested andvalidated by means of a complete parametric
identi-fication using both numerical simulations and experi-mental
data:
Cl(t) = a0 + a1ϕ + a2ϕ̇ + a3|ϕ|ϕ̇ + a4|ϕ̇|ϕ̇(Ref. [1])
Table 1 The geometrical characteristics of several 80◦ deltawing
models
Model c [mm] b [mm] S/Swt
Ref. [6] 428 150 0.032
Ref. [7] 1760 620 0.041
Ref. [8] 422 149 0.085
Ref. [11] 200 70 0.019
Refs. [12, 13] 479 169 0.006
Cl(t) = a0(ϕ) + a1(ϕ)ϕ̇ + a2(ϕ)ϕ̇2 + a3(ϕ)ϕ̇3+ a4(ϕ)ϕ̇4 (Ref.
[11])
Cl(t) = a1ϕ + a2ϕ̇ + a3ϕ3 + a4ϕ2ϕ̇ + a5ϕϕ̇2(Ref. [14])
Cl(t) = a0ϕ + a1ϕ̇ + a2|ϕ̇|ϕ̇ + a3ϕ3 + a4ϕ2ϕ̇(Refs. [12,
13])
Accurate modeling of wing rock is essential to designcontrol
systems able to suppress or alleviate this formof degraded
stability. This paper tries to contributeto this field, providing a
comprehensive parametricmodel for wing rock dynamics of a 80◦ delta
wingconfiguration, with and without slender forebody.
2 Parametric model
The considered analytical model was derived and ex-perimentally
validated in [12, 13]. The nonlinear dif-ferential equation (single
degree of freedom roll dy-namics) which describes the free motion
of the rollangle ϕ is
ϕ̈ + a0ϕ + a1ϕ̇ + a2|ϕ̇|ϕ̇ + a3ϕ3 + a4ϕ2ϕ̇= ϕ̈ − Ĉl(ϕ) = 0
(1)
where a0, a1, a2, a3, a4 are the parameters relativeto the
experimental conditions (i.e., angle of attack,Reynolds number, and
wing characteristics). The timederivatives are nondimensional (the
time scaling factoris b/2V ).
Note that
Ĉl(ϕ) = qSbIxx
· Cl(ϕ) (2)
is the normalized rolling moment coefficient, i.e., theexternal
driving torque.
3
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Fig. 1 The 80◦ delta wingmodel configurations:model A (wing),
model B(wing + nose tip), model C(wing + forebody + nosetip), model
D (wing +forebody)
The restoring moment a0ϕ + a3ϕ3 exhibits a typ-ical trend with
softening of linear stiffness a0 (Duff-ing equation). As a
consequence the system is stati-cally divergent for ϕ >
√−a0/a3. The damping co-efficient (a1 + a4ϕ2) is nonlinear and
negative forϕ <
√−a1/a4 (Van der Pol equation). The system isdynamically
unstable for lower roll angles becomingstable as ϕ increases up to
the inversion point. The co-ordinate for this dynamic stability
cross-over is not co-incident with limit cycle amplitude, as the
stability offinal state occurs when
E ≡∮
�
Ĉl(ϕ) dϕ = 0 (3)
This condition is required for the balance between dis-sipation
and generation of energy E and for a stableoscillatory limit cycle.
Dynamic stability and limit cy-cle characteristics are also
influenced by the additionaldamping produced by the term a2|ϕ̇|ϕ̇.
The equiva-lence of reduced order models with the
experimentalsystem is discussed in detail in [12, 13].
3 Experimental activity
Free-to-roll experiments were performed on a set ofdelta wing
models (see Fig. 1).
The first set of experiments was performed onmodel A for α =
21◦–45◦, V = 15 m/s–40 m/s, Re =486000–1290000, and ϕ0 = 0◦–90◦
(ϕ̇0 = 0). Theseresults were presented in [12, 13]. Additional
experi-mental data were obtained on the complete set of mod-els A,
B, C, and D for α = 25◦–45◦, V = 30 m/s,Re = 950000 and ϕ0 = 20◦
(ϕ̇0 = 0). A special setof measurement was performed on model C for
α =27.5◦ and α = 32.5◦ with variable airspeed V = 15–40 m/s and ϕ0
= ±90◦ (ϕ̇0 = 0). These last results arediscussed in the present
paper.
The experimental tests were carried out in the D3Mlow speed wind
tunnel at Politecnico di Torino. Thetest section is circular (3 m
in diameter). The turbu-lence level is 0.3 % at V = 50 m/s.
The model was a 80◦ delta wing with sharp lead-ing and trailing
edges, made in aluminum alloy. Sharpleading edge delta wings
aerodynamics exhibit a min-imal sensitivity to the effects of
Reynolds number as
4
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Fig. 2 The experimentalsetup
the separation of wing primary vortices is fixed alongthe
leading edge. The results presented in [12, 13]confirm this
insensitivity for model A. The dimen-sions are: root chord c = 479
mm, span b = 169 mm,thickness 12 mm, and bevel angle 20◦. Four
differentmodel configurations were obtained by adding a
cylin-drical forebody (length 84 mm) and a conical nosetip (length
90 mm), as explained in Fig. 1. The over-all length of the complete
model with fuselage andnose tip (model C) is a 568 mm. A set of
experimentswas performed with a modified version of model Cwith two
symmetrical forebody strakes (installed with−30◦ negative
dihedral). The wing longitudinal bodyaxis and the bearings axis
coincide. The rotating sys-tem was statically balanced. Note that
feature is notverified in some of the previous experiments
availablefor reference.
The C-shaped support (Fig. 2) was mounted on avertical strut
which was able to rotate so that the an-gle of attack could change
while the model centroidremained at the center of the test
section.
The model was connected to a horizontal shaft sup-ported by
rolling bearings. In order to minimize thefriction of the angular
transducer, the motion of thewing was measured by an optical
encoder, linked withthe rotating shaft using an elastic joint
without back-lash. This digital transducer was able to provide a
res-olution of 0.45◦/bit.
A pneumatic brake was adopted to keep the wingin the initial
angular position. During wind on runs,a trigger signal was sent by
the operator to the data ac-quisition unit and the model was
released by a pneu-matic cylinder fit inside of the vertical arm of
the C-shaped support.
The digital signals generated by the encoder, whichidentify the
sign, the increment and the zero crossingof ϕ(t), were conditioned
by an electronic device con-sisting of an incremental counter and a
12 bit digital to
analog converter. Both the analog output and the zerocrossing
trigger signal were multiplexed with a rate of50 samples/s over a
period of 45 s. The data acqui-sition system was based on a 12 bit
analog to digitalconverter and an oscilloscope for the real time
signalmonitoring.
The amplitude and the oscillation frequency of thelimit cycles
were identified after the numerical elab-oration of the time
histories ϕ(t) with a spectral ana-lyzer. The angular rates were
evaluated numerically.
The rolling moment coefficient was evaluated con-sidering
that
Cl = IxxqSb
ϕ̈ (4)
where Ixx = 0.0008738 Kg m2 is the moment of in-ertia about the
roll axis for model A and Ixx =0.0008896 Kg m2 for model C.
The coefficient Cl includes the effect of friction(Clf ):
Cl = Claer + Clf = Claer + Cl0f + μf ϕ̇ (5)where Claer is the
aerodynamic rolling moment coeffi-cient. Friction was neglected
taking into account thatthe limit cycle parameters (Δϕ,k) for model
A mea-sured at TPI are very similar to those presented in [8]for
comparable test conditions (see Fig. 3). The exper-imental setup
adopted by Arena and Nelson is defi-nitely frictionless as the
rotating shaft is supported byair bearings. Therefore, these data
are taken as a ref-erence to estimate the impact of friction on TPI
mea-surements. The trend of reduced frequency is coinci-dent but
shifted to higher values. This difference isa direct consequence of
the different rotational iner-tia of the experimental apparatus
adopted in [8]. Theexperiments performed by Arena and Nelson
estab-lish that the oscillation frequency is proportional to1/
√Ixx and that the amplitude Δϕ is not substantially
5
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Fig. 3 The experimental limit cycle characteristics (Δϕ is the
oscillation amplitude in roll and k = πf b/V is the reduced
oscillationfrequency) for several 80◦ delta wing models at
different angles of attack α
changed by Ixx . Anyway, the inertial scaling of theresults
performs correctly with a perfect overlap, asconfirmed by the
comparison presented in [12, 13].As a further comment, the tests
performed by Hanffin [15] on a free-to-roll apparatus similar to
the TPIrig demonstrate that only the constant friction termCl0f
is required to model the system friction (if re-quired),
regardless of the angular velocity and loadsacting on the wing. A
direct measurement of the break-out torque due to friction in wind
off conditions con-firmed that this term is very small for the TPI
oscilla-tory rig (L = 4.5 · 10−4 Nm equivalent to 0.1 % of
theaveraged aerodynamic term for V = 30 m/s). It mustalso be
observed that the wing oscillations were al-ways immediately
triggered as the pneumatic brake fitinto the TPI support system was
released (at least formodel A), even for ϕ0 ≈ 0. A very interesting
analy-sis of the effects of model axis of rotation and frictiondue
to bearings on wing rock experimental data is alsogiven in
[16].
In [17, 18], an extensive derivation of criteria forinertia
similitude between different models, or modeland aircraft, is
given. These criteria state that simili-tude is ensured when the
two configurations possessthe same nondimensional ratio Ixx/ρb5.
The nondi-mensional inertias for different models and aircraft
arecompared in [12, 13]. This analysis demonstrates thatrelevant
scaling factors are required in order to com-pare in-flight wing
rock with free-to-roll experiments.
Similar factors apply for models with the same geom-etry tested
in different wind tunnels.
4 Results
The wing rock phenomenon (see Fig. 4) becomes sta-ble after a
build up phase. These oscillations are sus-tained around a state at
which the energy generation atlower amplitudes and the dissipation
at larger ampli-tudes are balanced. The build up phase for model
A(basic winged model configuration for α = 30◦) ischaracterized by
a very rapid increase of oscillationamplitude with fast convergence
to limit cycle. Differ-ently, the build up phase for model C
(complete wing-body configuration) is very progressive with a
longertransient. After that intermediate phase, the final stateis
finally reached. The plot of the experimental resultsshows that the
center of the elliptic cycles is shiftedfrom the origin (Δϕ ≤
2.5◦). This asymmetry is a con-sequence of the support
interference. Static flow visu-alizations on model A [12, 13]
confirmed that the wingvortices were slightly displaced even for ϕ
= 0◦.
The effect of model configuration on the final stateparameters
(amplitude and reduced frequency) for in-creasing angles of attack
is presented in Fig. 5. Thepresence of the fuselage induces a
reduction of oscil-lation amplitudes as if (at least apparently)
the aero-dynamic damping of the system was increased with
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respect to the wing only configuration (model A).All
configurations exhibit an increase of amplitudesΔϕ followed by a
sharp reduction for larger anglesof attack, due to the presence of
vortex breakdown
Fig. 4 Wing rock oscillatory dynamics: build up and limit
cycle(models A and C)
on the wing (stabilizing effect canceling the hystere-sis on
vortex displacements that drives wing rock dy-namics). Stable limit
cycles are still observed up toα = 45◦ (model A). Differently, the
oscillations arecompletely suppressed for α ≥ 45◦ for the other
mod-els equipped with fuselage. Another interesting featureis the
scaled similarity for the trend of amplitudes ofmodels A and B
(wing-body configuration with nosecone only). This point suggests
that the presence of thecone apex alters the damping (scaled limit
cycle am-plitudes) while the presence of the forebody is a
dom-inant factor with a major nonlinear effect for the
char-acteristics of limit cycle (models C and D), generatedby the
asymmetric behavior of forebody vortices. As amatter of fact, the
vortices generated on the apex of thefuselage for model B are
immediately interacting withthe primary wing vortices without
developing asym-metric α-dependent patterns, typical of slender
fore-bodies (models C and D). Model C shows an interest-ing
singularity for α = 27.5◦ as wing rock oscillationsare not
triggered (see Fig. 14). The trend of oscillationamplitudes for
model D shows that the suppressionof wing rock at higher angles of
attack is anticipated:the wake disturbances generated by the blunt
fuselageapex promote—through their interference—the break-down of
wing vortices (stabilizing effect). The pres-ence of the fuselage
also scales down the limit cyclereduced frequencies. This is a
consequence of the al-teration of aerodynamic damping brought into
the sys-tem by the additional forebody vortex dynamics
super-imposed to wing vortices. The onset of breakdown forwing
vortices triggers a sharp frequency increase for
Fig. 5 Experimental data: effect of model configuration on
amplitude and reduced frequency (models A, B, C, and D—V = 30
m/s,Re ≈ 950000)
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Fig. 6 Experimental data: amplitude and reduced frequency (model
C—V = 30 m/s, Re ≈ 950000)
Fig. 7 Experimental data: effect of initial conditions on
amplitude and reduced frequency (model C—V = 30 m/s, Re ≈
950000)
the basic winged configuration (model A) that is lessevident for
the other models.
In Fig. 6, the limit cycle characteristics for model Care
presented for V = 30 m/s and ϕ0 = 20◦. The limitcycle is stable for
lower angles of attack only, and forα ≥ 37.5◦ (wing vortex
breakdown starts to occur) ei-ther the oscillation amplitude
fluctuates or the motiondisappears completely. As a matter of fact,
the type ofroll dynamics that is observed for the complete
config-uration is only partially described as a stable
ellipticallimit cycle. The reduced frequency does not exhibit
aspecific trend as a response to wing vortex breakdown(differently
from model A) with a very large scatter forhigher α.
The effect of initial conditions on the limit
cyclecharacteristics for model C (α = 27.5◦ and α = 32.5◦)is
presented in Fig. 7. No wing vortex breakdown isobserved for ϕ0 =
0◦. The limit cycle is unaffected by
the initial release roll angle ϕ0. A similar result wasfound for
model A in [12, 13]. The unique singularityis for α = 27.5◦ as
several tests confirmed that wingrock is not triggered for initial
conditions ϕ0 ≈ 0 andV = 30–35 m/s. The steady state is a small
ampli-tude wing vibration in roll and the spectral frequencyis
still very close to the one of the limit cycle found forlarger
initial conditions. An explanation based on theanalysis of the
analytical model is presented later. Rolldivergence or spinning are
not seen in experiments formodels A and C, since above a certain
angle of attack(or alternatively for larger roll angles), vortex
break-down appears on the wing and it contributes a damp-ing moment
that reduces the steady state amplitude.Therefore, limit cycles are
seen instead of divergence.The observation of roll divergence in
wind tunnel ex-periments is also strongly affected by effective and
vir-tual dihedral (i.e., the angular measure of lateral sta-
8
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Fig. 8 Experimental data: effect of forebody strakes on
amplitude and reduced frequency (model C—V = 30 m/s, Re ≈
950000)
bility) for the model tested. Virtual dihedral is mainlychanged
by leading edge bevel angle, sting and modelshapes. Another
contributing element is the position ofthe axis of rotation that
should be coincident with theinertial axis.
Static surface flow visualizations on model C(mini-tufts were
glued on the upper part of the wing)show that, for α > 35◦, the
wing vortices are led toasymmetry by the interference with forebody
wakeand vortical patterns.
The effect of forebody strakes on model C oscilla-tory response
is presented in Fig. 8. The position andthe size of the strakes was
selected according to anempirical review of existing solutions [3].
These non-lifting devices force the forebody vortices to
becomesymmetric neutralizing their natural tendency to asym-metry.
They also affect the interference with the liftingsurface. The
oscillation amplitude Δϕ exhibits a be-havior that is very similar
to model A for α ≤ 35◦.The wing rock oscillation disappears
completely forα > 35◦ due to the anticipation of vortex
breakdowninduced by the interaction of forebody strakes with
theflow emanating from the wing leading edge. The sin-gularity for
α = 27.5◦ is completely canceled and astable limit cycle is reached
even for ϕ0 ≈ 0. The re-duced frequency is marginally affected. As
expected,the strakes deflect and enforce the symmetric align-ment
of forebody vortices. These effects are tuned se-lecting their
dihedral angle and their position along theforebody.
The parameters ai (Fig. 9) were identified for mod-els A and C
by means of least-squares approximationof the experimental results
with coherent test condi-
tions. It can be said that, for α ≤ 35◦, the coefficientsa0 and
a3 representing the restoring action (stiffness)are similar for
both configurations while the coeffi-cients a1, a2, and a4
representing the damping actiondiffer substantially. As a matter of
fact, the presenceof the fuselage alters the hysteresis mechanism,
i.e.,the damping of the system. For α > 35◦, no
furtherconsideration can be derived for the trend of the
coef-ficients of models A and C. The only remark is that
thecoefficients for model A (see Fig. 10) still reproducethe
oscillatory response (amplitude Δϕ and reducedfrequency k) with
accuracy in the complete α-rangewhile the simulated response for
model C fails to com-ply with experiments for α > 35◦ (see Fig.
11). Thecoefficients of the analytical model are obtained
fromnumerical fit of experimental data derived with
initialconditions internal to the final state, i.e., the
extensionof its validity to larger roll angles 60◦ < Δϕ < 90◦
isnot straightforward. As a matter of fact, the dynamicstability of
trajectories with large angular perturba-tions should be
investigated through other experimen-tal methods (large amplitude
direct forced oscillationtechniques as an example [15]).
The steady state offset of limit cycle, measuredfrom experiments
and reproduced by the analyticalmodel, is presented in Fig. 12. The
experimental datafor model A show a moderate constant offset dueto
support interference (not shown by the analyticalmodel that is
unable to reproduce asymmetric oscil-latory cycles). The data for
model C demonstrate thatthe complete configuration oscillates with
an angularoffset increasing up to Δϕ = 20◦ (an effect of asym-metry
of forebody vortices). As expected, the analyt-
9
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Fig. 9 Analytical model: fitting of experimental data (models A
and C—V = 30 m/s, Re ≈ 950000)
ical model filters the presence of the offset with aunique
exception for α = 42.5◦ (see Fig. 16) that is anonoscillatory
condition, i.e., the solution is attractedby the restoring actions
providing a stable asymmetrictrim.
A detailed analysis of the terms of the analyticalmodel is given
in Fig. 13 for model A at α = 30◦. The
damping coefficient (a1 +a4ϕ2) is nonlinear and nega-tive for ϕ
<
√−a1/a4 (Fig. 13). The system is dynam-ically unstable for lower
roll angles becoming stable asϕ increases up to the inversion
point. The condition forequilibrium (limit cycle) is obtained as
the balance be-tween dissipation and generation of energy.
Dynamicstability and limit cycle characteristics are also
influ-
10
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Fig. 10 Comparison of experimental data and model fitting:
amplitude and reduced frequency (model A—V = 30 m/s, Re ≈
950000)
Fig. 11 Comparison of experimental data and model fitting:
amplitude and reduced frequency (model C—V = 30 m/s, Re ≈
950000)
Fig. 12 Comparison of experimental data and model fitting:
steady state offset (models A and C—V= 30 m/s, Re ≈ 950000)
11
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Fig. 13 Restoring and damping actions as modeled in the
analytical model (model A—α = 30◦, V = 30 m/s, Re ≈ 950000)
enced by the additional damping produced by the terma2|ϕ̇|ϕ̇,
that shifts the transition angle from unstable tostable damping.
The reduced order model
ϕ̈ + a0ϕ + a3ϕ3 = 0 (6)describes an undamped system with
nonlinear stiff-ness. The restoring moment a0ϕ + a3ϕ3 exhibits
atypical trend with softening of linear stiffness a0. Asa
consequence, the system is statically divergent forϕ >
√−a0/a3 (unstable trim point).Model C shows an interesting
singularity for α =
27.5◦ as wing rock oscillations are not triggered (seeFig. 14).
After the release of the brake, model C con-verges to a
nonoscillatory steady state without a buildup phase. The one
distinctive feature of model C withrespect to models A, B, and D,
is that only model Chas both nose and forebody. Since none of the
othermodels exhibit anything resembling the peculiar be-havior of
model C, it would seem that the character-istic behavior of this
model would be strongly relatedwith the distinctive features it
contains. Therefore, itwould seem that the interaction between the
nose andforebody shed vortices, either exclusively or in
combi-nation with shed wing vorticity, can explain this un-usual
behavior. It is the interference between nose,forebody, and wing
vortices that cancels out the hys-teresis for the wing vortex
normal displacements, theprimary source of dynamic instability of
the system(this singular behavior for α = 27.5◦ is canceled bythe
adoption of forebody strakes that alter the interfer-ence between
forebody and wing vortices, delayed tohigher angles of attack). The
analysis of the equivalentanalytical model explains the situation.
The restoring
torque is positive for lower roll angles Δϕ ≤ 30◦ as ex-pected,
but the damping coefficient is always unstable(differently from the
situation presented in Fig. 13 formodel A). The level of negative
unstable damping forϕ ≈ 0 and ϕ̇ ≈ 0 is not sufficient to trigger
the build upphase as the restoring effect is prevalent. This is
con-firmed by the experiments that did not exhibit wingrock
oscillations for −15◦ < ϕ0 < 15◦ only. Largerinitial
conditions lead to large-amplitude limit cycleoscillations (see
Fig. 14). The analytical model pre-dicts oscillatory convergence
for |ϕ0| < 8◦ while failsto match the experimental data for |ϕ0|
> 8◦ (para-metric refit based on experimental data obtained
for|ϕ0| > 20◦ is required). Numerical experiments withthe
analytical model show that for ϕ ≈ 0 minimal in-crements of the
initial roll rate (prospin incrementsΔϕ̇ > 0.01 rad/s) lead to
divergence from symmetricflight. Here, it appears that there are
more stable limitcycles than one: Different initial conditions can
leadto at least two different motions, one small-amplitudeand one
large-amplitude limit cycle. If there are twostable limit cycles,
then there is an unstable limit cy-cle between them (as found by
the divergent responseof the analytical model for increasing
initial rate aboutϕ ≈ 0). Therefore, the basin of attraction for
the smallamplitude limit cycle is very narrow. Note that the
pa-rameters identified for |ϕ0| < 20◦ are unable to predictthe
large-amplitude attractor. The presence of sometype of switching in
the aerodynamics of the wingmay explain the need for a scheduling
of the parame-ters of the analytical model (critical state as
suggestedby [4]).
12
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Fig. 14 Comparison of experimental data and analytical model
response (model C—α = 27.5◦, V = 30 m/s, Re ≈ 950000)
A mismatch between experiments and analytical
model for the configuration C at α = 37.5◦ is pre-sented in Fig.
15. The experimental data demonstrate
that, due to forebody vortex asymmetry, the axis of
the limit cycle is shifted out bound at ϕ ≈ +19◦. Thedamping of
the system, as identified by the analyti-
cal model, is unstable for the complete ϕ-range. The
level of instability for ϕ ≈ 0 is sufficient to trigger thebuild
up phase and the motion is attracted by a stati-
cally stable trim point far from the condition of sym-
metry for the wing (unbalanced roll angle ϕ ≈ +19◦).The cubic
form of the restoring moment a0ϕ + a3ϕ3
13
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Fig. 15 Comparison of experimental data and analytical model
response (model C—α = 37.5◦, V = 30 m/s, Re ≈ 950000)
introduces into the model a second artificial trim pointfor ϕ ≈
−19◦ (statically stable), enforcing a symmet-ric behavior not
present in the real case. Therefore, thetwo attractors design the
trajectory of the limit cycledescribed by the analytical model in a
symmetric way.The dynamic instability drives the transition of the
os-
cillation from one trim point to another. This discrep-ancy can
only be corrected by adapting the form of therestoring torque to
the effect of oscillation offsets.
The same situation is observed for model C atα = 42.5◦ as shown
in Fig. 16. For this angle of at-tack, the wing rock is not
triggered and the limit cycle
14
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Fig. 16 Comparison of experimental data and analytical model
response (model C—α = 42.5◦, V = 30 m/s, Re ≈ 950000)
is not recognized during the experiments. Simulatingthe response
of the system with the analytical modelprovides convergence either
to positive or negative rolloffset, according to the sign of
initial conditions. Thelevels of dynamic instability are not
sufficient to drivethe switch between the two trim points
(stabilizing ef-fect induced by wing vortex breakdown). Once
again,the analytical model present a situation of symmetrythat is
artificial.
It must be underlined that when the analyticalmodel fails to
reproduce the wing rock response formodel C at α = 37.5◦,42.5◦, and
45◦ the restoringtorque is characterized by the presence of two
sepa-rate symmetric stable trim points for ϕ = 0. On thecontrary,
when for lower α, the stable trim point is atϕ = 0, the analytical
model correctly reproduces theamplitude and the reduced frequency
of the limit cy-cle, but it cannot model any offset of the cyclic
trajec-tory. As a conclusion, the function a0ϕ + a3ϕ3 cannot
fit or model asymmetric flight states as those inducedby
forebody vortex asymmetries.
Starting from the analytical model, the sign of stiff-ness and
damping aerodynamic terms is obtained (seeFig. 17). Model A
exhibits a uniform separation ofroll angle ranges for dynamic
stability that is the driv-ing mechanism of wing rock dynamics.
Note that theamplitude of the limit cycle falls in the
intermediaterange of the separation lines (i.e., where static
conver-gence and dynamic stability coexist). A wide area ofstatic
divergence is predicted for large roll angles. Aspreviously
discussed, divergence was never observedduring the experiments. The
analytical model actuallyfails to represent the restoring torque
contribution forlarger roll displacements. Model C shows a more
com-plicated pattern with a less uniform distribution of
therelevant areas. Static divergence is still found for verylarge
roll angles. The statically stable trim points withoffset found for
higher angles of attack are marked onthe diagram. The range for
dynamic stability is quite
15
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Fig. 17 Analysis of stiffness and damping aerodynamic
torques
extended for higher attitudes shrinking the amplitudeof the
oscillatory limit cycle for α > 40◦. The modeloverestimates the
extension of the dynamically unsta-ble region for α ≈ 27.5◦. This
explains the inability ofthe model to reproduce the large-amplitude
limit cycleoscillation observed during the experiments. Finally,the
area compatible with the small-amplitude limit cy-cle is defined in
compliance with the numerical simu-lations performed with the
analytical model.
5 Concluding remarks
A complete set of free-to-roll wind tunnel experimentshas been
performed on a 80◦ delta wing, with andwithout a modular fuselage
kit. A special set of exper-iments has been devoted to the
understanding of theeffect of airspeed and initial conditions on
limit cyclecharacteristics of model C (complete model with slen-der
forebody).
The results for the basic model (model A) confirmthe previous
set of data presented in [12, 13].
The experiments on the complete model with slen-der forebody
(model C) outline a relevant effect of an-gle of attack on limit
cycle characteristics, as for somemodel attitudes wing rock is not
triggered or even sup-pressed. The explanation is a more
complicated flowpattern, including the forebody vortices as a
drivingmechanism of interference with the wing vortices
andpromoting vortical asymmetries for symmetric flight,never
observed for the basic winged model (model A).
The effect of airspeed (Reynolds number) is margin-al and limit
cycle parameters (amplitude and reducedfrequency) are unchanged as
airspeed is varied withinthe limits used for the present testing
activity. A sin-gular behavior was observed for model C at α =
27.5◦and V = 30–35 m/s, where wing rock oscillations
areunexpectedly not triggered.
The initial release roll angle does not affect thelimit cycle of
model A (the limit cycle is a stable
16
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unique attractor), changing the build up transientphase only.
Differently, the complete model (model C)exhibits an occasional
sensitivity to the initial condi-tions ϕ0, precluding the build up
of the oscillations.When the motion is triggered, the limit cycle
charac-teristics still remain unaffected.
The comparison of experimental data shows thatthe presence of
the fuselage alters the damping termas observed by the decrease of
final state amplitudesand the increase of oscillation
frequency.
Static surface flow visualizations on model C(mini-tufts were
glued on the upper part of the wing)show that, for α > 35◦, the
wing vortices are led toasymmetry by the interference with forebody
wakeand vortical patterns.
The analytical model derived and successfully val-idated for
model A in [12, 13] was here applied tothe complete model case. The
analytical model com-plies with the experimental oscillation time
historiesmeasured on model C for α ≤ 35◦, while for higherangles of
attack the presence of forebody vorticeschanges substantially the
shape of the function de-scribing the restoring moment (the
softening formu-lation a0ϕ + a3ϕ3 adopted in the differential
equationdescribing wing rock roll dynamics). Attempts to cor-rect
the formulation did not produce a complete solu-tion for this
problem.
Acknowledgements The author wishes to acknowledge theinvaluable
technical assistance given by Mr. Andrea Bussolin.
References
1. Hsu, C.H., Lan, C.E.: Theory of wing rock. J. Aircr.
22(10),920–924 (1985)
2. Malcolm, G.N.: Forebody vortex control. Prog. Aerosp.Sci.
28(3), 171–234 (1991)
3. Cooperative programme on dynamic wind tunnel experi-ments for
manoevring aircraft. NATO AGARD AdvisoryReport 305 (1996)
4. Nelson, R.C., Pelletier, A.: The unsteady aerodynamics
ofslender wings and aircraft undergoing large amplitude ma-neuvers.
Prog. Aerosp. Sci. 39, 185–248 (2003)
5. Katz, J.: Wing-vortex interactions and wing rock.
Prog.Aerosp. Sci. 35, 727–750 (1999)
6. Levin, D., Katz, J.: Dynamic load measurements with
deltawings undergoing self induced roll oscillations. J.
Aircr.21(1), 30–36 (1984)
7. Nguyen, L.T., Yip, L.P., Chambers, J.R.: Self induced
wingrock of slender delta wings. In: AIAA Atmospheric
FlightMechanics Conference, Albuquerque, NM (1981)
8. Arena, A.S. Jr.: An experimental and computational
inves-tigation of slender wings undergoing wing rock. Ph.D.
Dis-sertation, University of Notre Dame, IN (1992)
9. Arena, A.S. Jr., Nelson, R.C.: An experimental study ofthe
nonlinear dynamic phenomenon known as wing rock.AIAA Paper 90-2812
(1990)
10. Arena, A.S. Jr., Nelson, R.C.: Measurement of
unsteadysurface pressure on a slender wing undergoing a self
in-duced oscillation. Exp. Fluids 16(6), 414–416 (1994)
11. Yoshinaga, T., Tate, A., Noda, J.: Wing rock of delta
wingswith an analysis by the phase plane method. In:
AIAAAtmospheric Flight Mechanics Conference, Monterey, CA(1993)
12. Guglieri, G., Quagliotti, F.B.: Experimental observationand
discussion of the wing rock phenomenon. Aerosp. Sci.Technol. 1(2),
111–123 (1997)
13. Guglieri, G., Quagliotti, F.B.: Analytical and experimen-tal
analysis of wing rock. Nonlinear Dyn. 24(2), 129–146(2001)
14. Nayfeh, A.H., Elzebda, J.M., Mook, D.T.: Analytical studyof
the subsonic wing rock phenomenon for slender deltawings. J. Aircr.
26(9), 805–809 (1989)
15. Hanff, E.S.: Large amplitude oscillations.
AGARD-R-776(1991)
16. Konstadinopoulos, P., Mook, D.T., Nayfeh, A.H.: Subsonicwing
rock of slender delta wings. J. Aircr. 22(3), 223–228(1985)
17. Quast, T., Nelson, R.C., Fisher, D.F.: A study of high
al-pha dynamics and flow visualization for a 2.5 % model ofthe F-18
HARV undergoing wing rock. In: AIAA AppliedAerodynamics Conference,
Baltimore, MD (1991)
18. Quast, T.: A study of high alpha dynamics and flow
visu-alization for a 2.5 % model of the F-18 HARV undergo-ing wing
rock. M.S. Thesis, University of Notre Dame, IN(1991)
17