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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.

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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

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mailto:giorgio.guglieri@polito.it

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

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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.

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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

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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

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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-

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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-

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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-

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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)

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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]).

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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

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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

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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

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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

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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.

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