NONLINEAR STATIC AEROELASTIC AND TRIM ANALYSISOF HIGHLY FLEXIBLE AIRCRAFT

Zhang Chi∗ , Zhou Zhou∗ , Meng Pu∗∗School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, PR.China

Keywords: Geometric nonlinearity, Static aeroelasticity, Trim, Stability, All-wing aircraft

Abstract

The static aeroelastic responses, trim solutionsand stability analysis of a high aspect-ratio all-wing aircraft without aileron nor rudder is pre-sented. The basics of the computational frame-work is the coupling of a geometrically nonlin-ear 3D co-rotational double-spar structural modelwith a 3D non-planar vortex lattice aerodynam-ic model. Inertia relief is introduced for the firsttime to count for the effects of mass distributionon the static deformations of the unconstrainedflying aircraft. The aeroelastic deflections, liftdistributions, aerodynamic coefficients and theirderivatives, as well as the trim solutions and flightdynamic roots are assessed and compared withrigid, linear and nonlinear structural models. Re-sults show that the overall aeroelastic responses,trim solutions and flight dynamic characteristic-s are highly affected by structural flexibility, re-vealing the importance of taking into account thegeometric nonlinearity in flexible aircraft designand analysis.

1 Introduction

Current studies on high aspect-ratio wings[1, 2]show that geometric nonlinearity plays a domi-nant role, and the inclusion of geometric nonlin-earity is required for accurate aeroelastic predic-tions, even under normal flight conditions. Manyresearches[3, 4] give important guidelines andwarnings about what to expect in the presenceof geometrical nonlinearities and when these be-come significant in aeroelasticity.

Drela[5] modeled a complete flexible aircraftas an assembly of joined nonlinear beams. Patilet al.[6] and Zhang[7] studied the effects of struc-tural geometric nonlinearities on the aeroelastic-ity and flight dynamics of HALE aircraft. Thegeometrically-exact beam theory and ONERAaerodynamic model were used for structural andaerodynamic analyses, respectively. Arena etal.[8] adopted physics-based 3D parametric mod-els of flexible wings based on an exact kine-matic approach, giving an improved understand-ing of the nonlinear phenomena to the dynamicaeroelastic behaviors of flexible wings. Xie etal.[9, 10] investigated the aeroelastic character-istics of a metal single-spar wing under large de-formations with nonlinear finite element method(FEM) and the strip theory, 3D lifting line theoryand non-planar vortex lattice method (VLM).

In addition, trim analysis and flight stabilityassessment are essential aspects of aircraft de-sign. Studies by the Department of AerospaceEngineering at the Politecnico di Torino on asolar-powered UAV[11, 12] show that the trimcondition and the natural frequencies of flight dy-namics could be significantly affected by the in-herent flexibility and non-linear deformations ofthe wing, leading to an overall change in the w-hole aircraft’s aeroelastic performances.

In engineering, a typical wing usually con-sists of spars, ribs, skin and concentrated mass(engines, propellers, etc.), whose stiffness andmass are not uniformly distributed. However, thecommonly used single beam model lacks the ca-pability to incorporate more details of the com-plex geometries and distributions of mass and

1

ZHANG CHI , ZHOU ZHOU

stiffness, which limits their practical usages tosome extent.

In this paper, an effective and computational-ly efficient framework for static aeroelastic trimand stability analysis of flexible subsonic aircraftis presented, where the structural geometric non-linearity and non-planar effects of the aerody-namic loads are considered. The co-rotationalbased geometrically nonlinear double-spar FEmodel is employed for the wing structures. Apotential flow based 3D non-planar VLM is con-sidered for its advantages in computational effi-ciency. The structural and aerodynamic modelsare tightly coupled by 3D thin plate spline (TPS)method. Inertia relief is introduced for the firsttime to count for the effects of mass distributionon the static responses of the unconstrained air-craft in flight. The deformed shape and redis-tributed aerodynamic forces, and nonlinear trimsolutions are obtained from a loosely-coupled it-eration procedure.

Considering a flexible high aspect-ratio all-wing aircraft, the aeroelastic responses, aerody-namic coefficients and derivatives, trim solutionsand flight dynamic roots are obtained and critical-ly compared for rigid, linear and nonlinear struc-tural models. The results give insights into the ef-fects of the structural geometric nonlinearities onthe aeroelastic responses, aerodynamic character-istics, trim conditions and flight dynamic charac-teristics of a complete flexible all-wing aircraft.

2 Theoretical Methodologies

2.1 Co-rotation based geometrically nonlin-ear structural model

To capture the large, nonlinear structuraldeformations of the HALE aircraft, a co-rotational framework is developed based onEuler-Bernoulli beam elements.

In a co-rotational formulation, the rigid-bodymotions are separated from the strain-producingdeformations at the local element level. Hereinit is assumed that the ’internal element behavior’is linear whereas nonlinearity are introduced viathe co-rotational technique. This is accomplished

by attaching an element reference frame to eachelement and two nodal reference frames to thenodes of each element; these frames translate androtate with the displacements of the element.

The more detailed descriptions of 3D formu-lations for a spatial beam and its tangent stiffnessmatrix, as well as the numerical implementationof the rotational updates and the overall solutionstrategy can be found in the co-rotational nonlin-ear finite element literature[13, 14].

2.2 Non-planar VLM for aerodynamics

The aerodynamics of deformed lifting surfacesare computed by non-planar VLM, in which thelifting surfaces are deformed with the structure,the boundary condition and the locations of thecollocation points are updated according to thedeformed configuration of the wing. For moredetails the reader is referred to[15].

2.3 TPS for aerodynamic/structure coupling

As far as load transfer is considered, the thin platespline (TPS) based on the virtual work principleis chosen. In this way, it is possible to projecta force acting on the aerodynamic grid onto thestructural one. More thorough treatises on thistopic are also shown in[16].

2.4 Inertia relief

If a free body is accelerating due to constant un-balanced loads, the inertia relief solution pro-vides the ability to obtain static deflections rel-ative to a set of reference points attached to themoving coordinate system.

The basic equation of inertia relief is

KKKuuu = FFF −MMMaaa (1)

where uuu is displacements relative to the movingsystem and aaa is the steady accelerations to be de-termined from the mass and loads.

If ΦΦΦR is a matrix whose columns define therigid body motions of the structure, then for afreebody,

ΦΦΦTRKKKuuu = ΦΦΦ

TRFFF −ΦΦΦ

TRMMMaaa = 000 (2)

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NONLINEAR STATIC AEROELASTIC AND TRIM ANALYSIS OF HIGHLY FLEXIBLE AIRCRAFT

Start

Model initialization:

Double-beam structural FE model

Non-planar VLM model

Initial flight status

Inertial properties and center of mass

calculation

Aerodynamic loads computation

Follower force calculation

Inertia relief correction

TPS force transformation

CR based nonlinear structural static

analysis

Convergence

criteria

Results output

End

Aerodynamic coefficients

and derivatives calculation

Structural model update

Aerodynamic model update

No

Yes

Fig. 1 Static aeroelastic analysis flow chart

However, since the full-sized vector, aaa, is arigid body motion, it may be defined in terms ofaccelerations, aaaR, by the equation

aaa = ΦΦΦRaaaR (3)

Combining Eq.(3) into Eq.(2), we obtain

aaa = ΦΦΦRMMM−1ΦΦΦ

TRFFF (4)

where the total mass matrix for the reference co-ordinates is

MMM = ΦΦΦTRMMMΦΦΦR (5)

The resulting equation defined in Eq.(1) maynow be arbitrarily constrained since the total loadis balanced by the inertia forces.

2.5 Aeroelastic model and solution strategy

As illustrated in Fig.1, the nonlinear aeroelasticanalysis is conducted by iterative calculations oftwo modules, the aerodynamic analysis moduleand the structural static analysis module.

The procedure for the geometrically nonlin-ear static aeroelastic analysis starts with the ap-propriate aerodynamic and structural discretiza-tion and initialization, followed by the iterativecalculations. For each cycle of computation, theaerodynamic model and inertia properties of thewhole aircraft are updated according to the lat-est structural deformations. The external forces

Table 1 Geometrical and structural properties ofthe all-wing aircraft

Wing span 7.0 mWing surface area 3.7 m2

Chord length 1.2m/0.4m (IW/OW)Dihedral angle 0◦/6◦ (IW/OW)Incidence angle 8◦/6◦ (IW/OW)Front spar (IW) 30%c, 30 mm in diameterRear spar (IW) 45%c, 14 mm in diameter

Front spar (OW) 15%c, 24 mm in diameterRear spar (OW) 65%c, 14 mm in diameter

Young’s modulus 103.7 GPaPoisson’s ratio 0.27

Equivalent density 9,100 kg/m3

Total mass 14.7 kgPayload and batteries 4 kg

are transformed into the structurally equivalentnodal forces by TPS, and then corrected by in-ertia relief method. The stiffness matrices andstructural deformations are updated based on thelatest aerodynamic and non-aerodynamic forces.As one cycle finishes, the structural deformationsand aerodynamic characteristics will be evaluatedand tested for termination. If termination criteriaare not met, a new iterative cycle will be exciteduntil a converged solution is found.

3 Modeling of the flexible high aspect-ratioall-wing aircraft

The present model is a flexible high aspect-ratioall-wing aircraft. The inner wing (IW) and out-er wing (OW) each has two hollow tubular spars.The wing platform and wing structure are illus-trated in Table 1 and Fig.2, respectively.

As with Fig.2, the outer wing each has twospars and 10 equally spaced ribs that are mod-eled as beam elements to account for their effect-s on the mass and stiffness distributions of thewing. The properties of the engines and pylons,and batteries and payloads are also accounted foras concentrated mass.

Figure 3 shows the aerodynamic model of theaircraft and there are 28×10 vortex lattices.

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ZHANG CHI , ZHOU ZHOU

Fig. 2 Structural finite element model

Fig. 3 Aerodynamic model

Fig. 4 Bending deflections of the wings underdifferent angles of attack (H=500m, V=14m/s)

Fig. 5 Top view of the initial and deformed shapeof the all-wing aircraft (α = 10◦)

4 Numerical Results

We now apply our aeroelastic and trim analysesto the flexible aircraft introduced in the previoussection. The cruise speed of the aircraft is set to14 m/s and the altitude is 500 m.

The structural responses of the wing structureare analyzed by the traditional linear model (rep-resented by the label ’linear’), and the proposedco-rotational based geometrically nonlinear mod-el (represented by the label ’nonlinear’), in whichgeometric nonlinearity effects brought by largedeformations are taken into account.

4.1 Bending deformations

Fig. 4 shows the variations of the static equilibri-um position of the wings, with α=-6◦, -2◦, 2◦, 6◦

and 10◦, respectively.It shows that the vertical displacements of the

wings keep increasing as α increases, and the ver-tical displacement at the wingtip goes to about40% of the semispan when α is 10◦. Accordingto the nonlinear results, the finite-span wing hasa large lateral displacement of about 10% of thesemispan. This is a typical nonlinear case of aflexible wing with large deformations. While theresults attained by the linear model indicate thatthe vertical displacement of the wingtip goes al-most linearly, little lateral displacement occurs.

Figure 5 is the top view of the initial and de-formed shape of the right wing, when α is 10◦.A lateral displacement in the nonlinear result isevident, which is in accordance with the analy-sis above. The linear result shows a small lat-eral displacement, which is caused by the lateralforces introduced by the wing’s bending deflec-tions. Comparing the deformed state with the ini-tial one, it can also be seen that, the wing bend-s upwards and it is accompanied by a backwardbending deformation. This phenomenon is fre-quently seen in high aspect-ratio wings.

4.2 Torsional deformations

The torsional deformations of the wings areshown in Fig.6 for selected angles of attack.

It can be seen that a positive torsion occurs

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NONLINEAR STATIC AEROELASTIC AND TRIM ANALYSIS OF HIGHLY FLEXIBLE AIRCRAFT

� � � � � � � � � �

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0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

- 4

- 3

- 2

- 1

0

1

Torsi

onal

angle

(°)

S p a n w i s e l o c a t i o n ( y / s e m i s p a n )

Fig. 6 Torsional aeroelastic deflections of thewings under different angles of attack

when α is low, while a negative torsion occurswhen α is high. This is because the front sparplacing at 15% chord has a larger diameter thanthe rear spar placing at 65%, hence it is muchstiffer than the rear one, making the wing’s e-lastic axis lie in front of the aerodynamic cen-ter (A.C.) line of the wing. When α is low, thelift on the A.C. is too low to balance the pitch-up moment existing on the A.C., and the nose-up moment brings the positive torsion. When α

goes high enough, the lift on the A.C. is high e-nough to counteract the invariable pitch-up mo-ment. When α increases further, a pitch-downmoment arises with the increased lift, bringingthe wing’s negative torsional deflections. It canbe inferred that this kind of structural design andtorsional deformation can prevent the static tor-sional divergence to some extent, but may causethe reversal effects like control reversal.

Besides, the effects of geometrical nonlinear-ity on the torsional deformations can be inferredfrom the results obtained by linear and nonlinearmodels. The torsional deformations obtained bythe nonlinear model are larger than that obtainedby the linear one.

4.3 Spanwise lift distributions

Figure 7 illustrates the spanwise distributions oflift coefficient of the wings, showing that OWhave higher lift coefficients than IW. It can also

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� � � � � � � � � � �

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

1 . 6

1 . 8

Lift c

oeffic

ient

S p a n w i s e l o c a t i o n ( y / s e m i s p a n )

Fig. 7 Lift coefficient distributions of the wingsunder different angles of attack

be seen that the flexible wings have higher liftthan the rigid one when α is low, and lower liftwhen α is high, and the flexibility effect increas-es with the increase of α for both the linear andnonlinear structural models. This is caused by t-wo main reasons, one is that in the present caseof large flatwise bending, the aerodynamic forcesnot act in the vertical direction. The other reasonis that the flexible wings undergo negative tor-sions when α is high, which decreases the localangles of attack, and then the lift drops.

Comparison of linear and nonlinear resultsshow the lift from the nonlinear model is lowerthan that from the linear one; it agrees with thephenomenon that the nonlinear model produceslarger negative torsions.

The initial and final aerodynamic load predic-tions solved by the nonlinear model, when α is10◦, are illustrated in Fig.8. It can be inferredthat the rigid or linear model will give inaccurateresults when the structural flexibility is sufficient-ly high to lead to a large deformation; in this casethe nonlinear models can provide more accuratepredictions.

4.4 Variations of aerodynamic characteris-tics vs. angle of attack

The variations of aerodynamic coefficients andtheir derivatives under different angles of attackare calculated using the non-planer VLM coupled

5

ZHANG CHI , ZHOU ZHOU

Fig. 8 Initial and final aerodynamic models ofthe flexible wing

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 20 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

1 . 6

Lift c

oeffic

ient

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 9 Variations of lift coefficient vs. AOA

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 24 . 1

4 . 2

4 . 3

4 . 4

4 . 5

4 . 6

4 . 7

4 . 8

4 . 9

5 . 0

Lift c

urve s

lope

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 10 Variations of lift curve slope vs. AOA

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2- 0 . 8

- 0 . 7

- 0 . 6

- 0 . 5

- 0 . 4

- 0 . 3

- 0 . 2

Deriv

ative

of pi

tch m

omen

t coe

fficien

t

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 11 Variations of Cmαderivative vs. AOA

with linear and nonlinear structural models, andcompared with the rigid case, as shown in Fig.9through Fig.14.

Figure 9 and Fig.10 show that the lift and liftcurve slope (CLα

) of flexible aircraft are all low-er than those of a rigid one, and results from thenonlinear model are even lower than those fromthe linear one. With α increases, the influenceof structural flexibility gets more conspicuous.When α is 10◦, the lift coefficients predicted bythe linear and nonlinear models are 1.17 and 1.12,which is only 82% and 79% of the lift of the rigidone, while the lift curve slopes drop from 4.59 to4.41 and 4.19, respectively.

Figure 11 shows the variations of pitch mo-ment coefficient derivative (Cmα

) obtained bythree different structural models. It can be seenthat the Cmα

s of flexible models are much lowerthan that of the rigid one. The nonlinear modelhas a smaller absolute value of Cmα

than the lin-ear one, and the difference increases with α.

Figure 12 shows the variations of longitudi-nal static stability margin with α. It can be seenthat the stability margin of rigid model increas-es almost linearly from 5.4% to 10.8% when α

increases from -6◦ to 10◦. Linear and nonlinearmodels show that the stability margin predictionobtained from flexible models is higher than whatfrom the rigid one. The variation and differencesamong different models is mainly affected by thecharacteristics of CLα

and Cmα.

Under aerodynamic loads, flexible wings un-

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NONLINEAR STATIC AEROELASTIC AND TRIM ANALYSIS OF HIGHLY FLEXIBLE AIRCRAFT

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 24

6

8

1 0

1 2

1 4

1 6

1 8Lo

ngitu

dinal

static

stabili

ty ma

rgin (

%)

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 12 Variations of longitudinal static stabilitymargin vs. angle of attack

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2

- 0 . 9

- 0 . 8

- 0 . 7

- 0 . 6

- 0 . 5

- 0 . 4

- 0 . 3

- 0 . 2

Deriv

ative

of sid

e-forc

e coe

fficien

t

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 13 Variations of Cyβvs. angle of attack

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2- 0 . 5 0

- 0 . 4 5

- 0 . 4 0

- 0 . 3 5

- 0 . 3 0

- 0 . 2 5

- 0 . 2 0

- 0 . 1 5

- 0 . 1 0

Deriv

ative

of ro

ll mom

ent c

oeffic

ient

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 14 Variations of Clβ vs. angle of attack

- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2- 0 . 0 6

- 0 . 0 5

- 0 . 0 4

- 0 . 0 3

- 0 . 0 2

- 0 . 0 1

0 . 0 0

0 . 0 1

Deriv

ative

of ya

w mom

ent c

oeffic

ient

A n g l e o f a t t a c k ( d e g )

R i g i d L i n e a r N o n l i n e a r

Fig. 15 Variations of Cnβvs. angle of attack

8 1 0 1 2 1 4 1 6 1 8 2 0

- 6

- 4

- 2

0

2

4

6

Trim

angle

of at

tack (

deg)

F l i g h t s p e e d ( m / s )

R i g i d L i n e a r N o n l i n e a r

Fig. 16 Variations of trim AOA with flight speed

8 1 0 1 2 1 4 1 6 1 8 2 0- 0 . 0 5- 0 . 0 4- 0 . 0 3- 0 . 0 2- 0 . 0 10 . 0 00 . 0 10 . 0 20 . 0 30 . 0 4

Trim

pitch

mom

ent c

oeffic

ient

F l i g h t s p e e d ( m / s )

R i g i d L i n e a r N o n l i n e a r

Fig. 17 Variations of trim Cm with flight speed

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ZHANG CHI , ZHOU ZHOU

dergo large bending deflections, which increas-es the equivalent angle of dihedral, the latitu-dinal and directional characteristics of the air-craft will change significantly. The derivativesof side-force, roll moment and yaw moment co-efficients at different angles of attack are plottedin Fig.13 through Fig.15, respectively. The side-force derivative Cyβ

is usually negative, and fre-quently small enough to be neglected entirely, butthe absolute values of Cyβ

of both linear and non-linear models increase with α, indicating that theaircraft gets more sensitive to the sideslip whenα is large, and so are the trends of Clβ and Cnβ

.Fig.14 shows that the absolute values of Clβ ob-tained by flexible models increase more rapidlythan that of rigid model, which will give rise tolarger frequency of Dutch roll.

As for the yaw moment derivative shown inFig.15, because the vertical stabilizer is too closeto the center of gravity, Cnβ

of rigid model isnegative but close to zero, showing an approxi-mate neutral weathercock stability, while the Cnβ

of flexible models are negative and decrease withα rapidly, indicating that the weathercock stabil-ity is unstable and goes worse with the upwardbending deflections.

4.5 Trim analysis

The static aeroelastic trim results for the all-wingaircraft is investigated. Here, the trim α is ob-tained by calculating the angle of attack that givesthe required lift.

Figure 16 indicates that the higher flightspeed is, the lower trim angle of attack is, andthe value of α required from a flexible aircraft ismore than that from a rigid one when flight speedis below 16 m/s. This is because the large flatwisebending makes the lifts do not act in the verti-cal direction. Large deformation and associatedloss of aerodynamic force in the vertical direc-tion leads to the requirement of a higher value ofα, and the value of α required from a nonlinearmodel is more than that from a linear one.

When the flight speed is above 16 m/s, thetrim angles of attack from the flexible models arelower than that from the rigid model, which is

caused by the positive torsional aeroelastic defor-mations when α is low.

The main significance of this result lies in thatdifferent models will give different stall speeds.The improper use of a model to predict the per-formance of a highly flexible aircraft may lead toinaccurate estimations of the flight envelope.

Figure 17 shows that the pitch moment intrim flight increases with flight speed. There isa pitch-down moment when the speed is below11 m/s, above witch a nose-up moment occurs.Besides, the absolute values of pitch moment intrim obtained from flexible models are larger thanthat from the rigid one. This result indicates thatstructural flexibility has effect on pitch momentin trim flight, and the differences of pitch mo-ment will have momentous effects on the designof elevators and longitudinal control laws.

Figure 18 shows that with an increase in flightspeed, the longitudinal static stability margin ofthe aircraft decreases, and is affected by struc-tural flexibility. The stability margin of flexibleaircraft is more sensitive than that of a rigid one.The stability margin of flexible models dropsfrom 11.4% to 5.8%, while it decreases merelyfrom 9.3% to 5.7% in rigid case, with the speedincreases from 8 m/s to 20 m/s, respectively.

4.6 Flight Stability Analysis

When flexibility effects are taken into account inflight dynamic analysis, the performances of aflexible aircraft are distinctly different from thoseof a rigid one.

Fig. 19 compares the flight dynamic roots ob-tained with different structural models. The rootsat a flight speed of 14 m/s are shown in Table 2.

Results show that the phugoid mode of theaircraft is unstable, and with flight speed increas-es, the spiral mode goes from stable to unstable.

The longitudinal and latitudinal modes are af-fected by wing flexibility. Since the large bend-ing deflection of the wing increases the aircraft’smoment of inertia about the pitch axis, the damp-ing of short period increases for the flexible mod-els, and the frequency decreases.

As for the latitudinal and directional modes,

8

NONLINEAR STATIC AEROELASTIC AND TRIM ANALYSIS OF HIGHLY FLEXIBLE AIRCRAFT

Table 2 Eigen roots of different structural models (V=14m/s)Mode Rigid Linear Nonlinear

Phugoid 0.050±1.005i 0.066±1.007i 0.062±1.006iShort Period -7.688±6.159i -6.975±5.100i -6.948±5.118i

Dutch -0.3190±0.860i -0.2794±1.099i -0.2769±1.105iSpiral -0.02573 -0.06908 -0.07001Roll -12.88 -13.09 -12.95

8 1 0 1 2 1 4 1 6 1 8 2 05

6

7

8

9

1 0

1 1

1 2

Long

itudin

al sta

tic sta

bility

margi

n in t

rim (%

)

F l i g h t s p e e d ( m / s )

R i g i d L i n e a r N o n l i n e a r

Fig. 18 Longitudinal stability margin vs. flight speed

due to the wing’s large bending deflections, theequivalent angle of dihedral increases, whichmakes Clβ increase, and leads to larger frequencyand smaller damping of Dutch mode. The spiraland roll modes of flexible models become morestable than the rigid one.

5 Concluding Remarks

Numerical simulations of the static aeroelasticdeformations and lift distributions, along with theaerodynamic coefficients and their derivatives, aswell as the trim and flight stability solutions of aflexible high aspect-ratio all-wing aircraft are p-resented. Comparisons for rigid, linear and non-linear structural models are also supplied.

Results show that the overall characteristic-s and trim solutions are significantly affected bystructural flexibility. Neglecting structural non-linearities can lead to very different predictionsof the aeroelastic behaviors of a flexible aircraft,which might be inaccurate and un-conservative.

- 2 0 - 1 5 - 1 0 - 5 00123456789

R i g i d L i n e a r N o n l i n e a r

Imag

inary

axis

R e a l a x i s

S h o r t P e r i o d

R o l l

- 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 20 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

R i g i d L i n e a r N o n l i n e a r

Imag

inary

axis

R e a l a x i s

D u t c h

S p i r a l

P h u g o i d

Fig. 19 Loci of the flight dynamic roots with anincreasing flight speed

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ZHANG CHI , ZHOU ZHOU

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