Journal of Sound and Vibration (1996) 193(4), 823–846

AEROELASTIC ANALYSIS OF A FLEXIBLEAIRFOIL WITH A FREEPLAY NON-LINEARITY

S.-H. K I. L

Department of Aerospace Engineering, Korea Advanced Institute of Science andTechnology, Taejon, Korea

(Received 10 January 1995, and in final form 10 October 1995)

A two-dimensional flexible airfoil with a freeplay non-linearity in pitch has been analyzedin the subsonic flow range. Structurally, the airfoil is modelled as finite beam elements andtwo spring elemens in pitch and plunge. A doublet lattice method is used for thetwo-dimensional unsteady aerodynamics to include the camber deflection effect. Thefictitious mass modal approach is adopted in order to use the consistent modal co-ordinatesfor the structures with non-linearity. Non-linear aeroelastic analyses for both the frequencydomain and time domain are performed for rigid and flexible airfoil models to investigatethe flexibility effect. Results are shown for models of different pitch-to-plunge frequencyratio. Responses involving limit cycle oscillation and chaotic motion are observed and theyare highly influenced by the pitch-to-plunge frequency ratio.

7 1996 Academic Press Limited

1. INTRODUCTION

Recently, many investigations for aeroelastic analysis have been made for systems withstructural non-linearities. Concentrated structural non-linearities such as those of freeplay,bilinear and cubic types, are known to have significant effects on the aeroelastic responsesof aerosurfaces. These non-linearities have significant effects even for small vibrationalamplitudes, and provide non-linear aeroelastic responses of four categories: dampedstable motion, periodic motion, chaotic motion and divergent flutter. Parameters affectingthese responses have been investigated in previous studies: initial conditions, frequencyratio and amount of non-linearity, etc. However, more studies are still needed tounderstand the non-linear aeroelastic phenomena comprehensively. A model with reduceddegrees of freedom is desirable for non-linear aeroelastic analysis, since the responses arecomplicated.

Aeroelastic analyses of a two-dimensional airfoil (or a typical section model) withconcentrated non-linearities have been performed by several investigators, because of itssimplicity and usefulness [1–11]. The typical section model gives a lot of insight andinformation about the aeroelastic phenomena. Previous studies on the two-dimensionalmodel usually used the rigid typical section model with a concentrated non-linearity inpitch. The aerodynamic loads were calculated by using the Theodorsen function in thefrequency domain or the Wagner function in the time domain; these are based on theassumption of chordwise rigidity. In the first non-linear aeroelastic analysis for an airfoilmodel, Woolston [1] and Shen [2] showed that the limit cycle oscillation may occur belowthe linear flutter boundary. McIntosh et al. [3] performed experimental work with a windtunnel model of two degrees of freedom. Lee [4] developed an iterative scheme for multiplenon-linearities using the describing function method and the structural dynamicsmodification technique. Yang and Zhao [5] studied the limit cycle oscillation of an airfoil

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0022–460X/96/240823+24 $18.00/0 7 1996 Academic Press Limited

.-. . 824

with pitch non-linearity subject to incompressible flow using the Theodorsen function.They made a comprehensive study of limit cycle flutter. Zhao and Yang [6] also studiedthe chaotic responses of an airfoil with cubic non-linearity in pitch and subject to steadyincompressible flow. Brase and Eversman [7] developed the transformation method whichconverts the harmonic aerodynamic force to a transient one, and found a jump responsein a two-dimensional model with pitch freeplay non-linearity. Hauenstein et al. [8, 9]made experimental and analytical analyses of an aerosurface with freeplay non-linearitiesin pitch and plunge degrees of freedom. They showed qualitatively good results betweenthe experiment and analysis. They concluded from the results that chaotic motion doesnot occur with the presence of a single non-linearity, but Price et al. [10, 11] pointed outthis is not true. They made a comprehensive aeroelastic study of a two-dimensionalairfoil with pitch non-linearities such as those of freeplay, cubic and bilinear stiffnesstypes.

The plunge-to-pitch frequency ratio (vh /va ) of previous models are usually less thanunity, which implies a large aspect ratio wing. The rigid chord assumption is valid for thelarge aspect ratio wing. However, the chordwise bending effect on the aeroelastic responseis significant when the aspect ratio of the wing becomes small, or when the mass ratio issmall. In order to understand the aeroelastic phenomena more comprehensively it isrequired to know the linear and non-linear aeroelastic response of a two-dimensionalflexible airfoil model. The chordwise bending effect of a cantilevered plate was investigatedin the supersonic range by Dugundji and Crisp [12]. Crisp [13] mentioned that the elasticchordwise bending effect in the typical section model may be an important factor in flutteranalysis. However, results of non-linear aeroelastic analysis for two-dimensional flexiblemodel have not yet been reported.

In this paper, a two-dimensional flexible airfoil with a freeplay non-linearity in pitchand subject to subsonic flow has been studied. A finite beam element and two springelements (one is non-linear) have been used for structural analysis. A doublet latticemethod (DLM) [14] has been used to accommodate the two-dimensional chordwisebending effect on the aerodynamic forces. The fictitious mass modal approach [15, 16] wasadopted in order to use the consistent modal co-ordinates for structures with non-linearity.Non-linear aeroelastic analyses for both the frequency domain and time domain have beenperformed for rigid and flexible airfoil models to examine the flexibility effect. For thefrequency domain analysis the describing function has been used with the V-g method. Themethod of Brase and Eversman [7] was used to obtain the transient aerodynamic forcesin the time domain. Results are shown for two models of different pitch-to-plungefrequency ratio (va /vh ). In the frequency domain, the limit cycle flutter speed andfrequency were obtained for various amplitude ratios. The time domain analysis wasperformed for various initial pitch amplitude ratios and free stream velocities. Responsesincluding the limit cycle oscillation and chaotic motion are observed and are shown to behighly affected by the pitch-to-plunge frequency ratio.

2. THEORETICAL ANALYSIS

2.1.

The structural non-linearity must be transformed to an equivalent linear system to solvethe non-linear aeroelastic problem in the frequency domain. The describing functionmethod used in this analysis is that which was used by Laurenson and Trn [17].

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Figure 1. The describing function of a freeplay non-linearity.

The freeplay non-linearity can be described by

{ f (a)}= 8Ka (a− s),0,

Ka (a+ s),

aq s−sQ aQ s

aQ−s 9, (1)

where a is the displacement, and s is the magnitude of the freeplay. Freeplay non-linearityis shown in Figure 1. The describing function for freeplay non-linearity d can be obtainedas

d=60,(1/p)[p−2a−2 sin 2a],

AE sAq s7, (2)

where A is the vibration amplitude, s is the freeplay gap and a=sin−1 (s/A). Then, theelastic force becomes f (A)= dKaA. This function is dependent on the amplitude ratio s/Aand the conventional linear flutter analysis algorithm can be applied when using thisfunction. Here the V-g method was used, which can predict the flutter speed of aperiod-one limit cycle oscillation. A time domain approach must be used to obtain thecharacteristics of the complicated motion.

2.2.

Generally, the aeroelastic analysis in the time domain is conducted on the generalizedmodal co-ordinate to reduce the computation time and memeory requirement. Innon-linear aeroelastic problems, structural properties vary as the displacement changes.Hence, using a constant set of normal modes from a fixed structural model gives inaccurateresults. To overcome this problem, Karpel proposed the fictitious mass method [15, 16].In this method, a large fictitious mass is added to the degree of freedom of mass matrixwhere structural change will occur. Then, the normal modes obtained from the freevibration analysis for the system with fictitious mass are used for the aeroelastic response.The basic idea of this method is that the local deformation due to a large mass enablesone to account for the structural changes.

The free vibration equation of motion of an n d.o.f. system with fictitious masses is

[M+Mf ]{u}+[K ]{u}= {0}, (3)

.-. . 826

Figure 2. A flexible beam airfoil model.

where [M] is the mass matrix, [K ] is the stiffness matrix and [Mf ] is the fictitious mass addedto the d.o.f. where structural change occurs (a list of symbols is given in the Appendix).The value of the fictitious mass is large enough not to induce numerical difficulty. Thisvalue is not so sensitive and can be selected from a simple model examination. Normalmode analysis for equation (3) gives a set of nf low frequency fictitious vibration modes[ff ]. Then the generalized mass and stiffness matrices are given as

[GMf ]= [ff ]T[M+Mf ][ff ], [GKf ]= [vf ]2[GMf ]= [ff ]T[K ][ff ], (4)

where [vf ] is the diagonal matrix of natural frequencies.A co-ordinate transformation is then performed to clean out the fictitious masses and

to form an actual basic case the stiffness matrix of which may differ from that of thenominal case by [DKb ]. The transformation is based on the natural frequencies [vb ] andeigenvectors [xb] associated with the equation of free undamped vibration in modalco-ordinates:

([GMf ]− [ff ]T[Mf ][ff ]){j� f}+([GKf ]+ [ff ]T[DKb ][ff ]){jf}= {0}. (5)

The mode shapes calculated for the fictitious mass finite element model are transformedto the basic case:

[fb ]= [ff ][xb ]. (6)

The basic case mode shapes [fb ] serve as a constant set of structural generalizedco-ordinates throughout the response analysis. More detailed explanation of the fictitiousmass modal approach are given in references [15, 16].

2.3.

The linear relationship between the aerodynamic force acting on the nodal point andthe vertical displacement of the nodal point is obtained as

{Fs}= q[Q]{us}, (7)

where [Q] is the aerodynamic influence coefficient calculated from the DLM. The structuraldisplacement is transformed into modal co-ordinate as

{us}=[fb ]{u� }, (8)

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where {u� } is the modal displacement. Then the generalized aerodynamic force can bewritten as

{F}=[fb ]T{Fg}= q[fb ]T[Q][fb ]{u� }= q[Q �]{u� }, (9)

where the underlined quantities are the modal co-ordinate quantities, and [Q �] is thegeneralized aerodynamic influence coefficient matrix.

Generally, the unsteady aerodynamic influence coefficient matrix is calculated for adiscrete reduced frequency k rather than calculated as a continuous function of the circularfrequency v. Thus the aerodynamic influence coefficient matrices should be approximatedas rational functions. There are many methods [18] of rational function approximation,but the Roger and Abel method [19] has been used here for simplicity and fast computationtime. The approximate form is

[Q(k)]= [A1]+ [A2](ik)+ [A3](ik)2 + sM+3

m=4

[Am ](ik)(ik)+ km

. (10)

Here the Ai’s are calculated from a least square fit. The km’s are constants to be determinedfor best fit. A simplex direct search method [20] is used to determine km for minimizingfitting error.

Figure 3. The natural frequencies of the flexible model: (a) the effect of torsional spring stiffness; (b) the effectof beam thickness. –––, Rigid; ——, flexible.

.-. . 828

Figure 4. The effect of the torsion to plunge frequency ratio on the flutter characteristics of a flexible typicalsection model (e/c=0·375, t/c=0·0128, vb/vh =3·74): (a) flutter speed; (b) flutter frequency. –––, rigid(present); ——, flexible; w, rigid (NASTRAN).

The transient aerodynamic force of an arbitrary wing motion can be written as

F� (t)= q0[A1]+ sM+3

m=4

[Am ]1u(t)+ q[A2]0 bUa1u(t)+ q[A3]0 b

Ua12

u(t)− q sM+3

m=4

[Am ]pmz� m ,

(11)

where pm =(b/Ua)km and z� m = ft0 u� (t) e−pm(t− t) dt.

The equation of motion for an aeroelastic system with concentrated structuralnon-linearity can be written as

[M]{u}+[C]{u}+ {R(u)}= {F}, (12)

where {R(u)} is the elastic restoring force, which is a function of displacement. Forpiecewise non-linearity, the restoring force can be written as

{R(u)}=[K]{u}+ { f (a)}, (13)

where [K] is the linear stiffness matrix without freeplay, and { f (a)} is the restoring forcevector, the elements of which are zero except for the non-linear element. For freeplay

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non-linearity, { f (a)} is given in equation (1). The procedure for applying the fictitious massmethod to freeplay is to generate the nominal modes with a nominal pitch stiffness (in thiscase zero), and then remove the pitch stiffness from the stiffness matrix and replace it bythe non-linear force vector.

Transformation of equation (12) into the modal co-cordinate ({u� }=[fb ]{u}) gives

[GM]{u}+[GC]u}+ {GR(u� )}=[F� ], (14)

where the generalized mass, damping matrix and restoring force vector are as follows:

[GM]= {fb}T[M]{fb}, [GC]= {fb}T[C]{fb},

[GR(u� )}=[GK]{u� }+ {fb}T{ f (a)}. (15)

Usually, the damping matrix [GC] is not calculated from the finite element formulation.Here we assumed the modal damping matrix to be given by

[GC]=2[z][va ], (16)

Figure 5. The effect of thickness on the flutter characteristics of a flexible typical section model (e/c=0·375,Ka =2400Nm/rad, va/vh =1·412): (a) flutter speed; (b) flutter frequency. –––, rigid; ——, flexible.

.-. . 830

Figure 6. The limit cycle flutter characteristics of flexible typical section model with a freeplay(Ka =2400Nm/rad, va/vh =1·412): (a) flutter speed; (b) flutter frequency. –––, D.F. (rigid); w, r, timesimulation (rigid); ——, D.F. (flexible); W, R, time simulation (flexible); — . —, flexible linear flutter; — . . —,rigid linear flutter.

where [z] is a diagonal damping coefficient matrix, usually taken with values from 0·005to 0·02, and [va ] is the natural frequency matrix of linear model.

The state variable and matrices are defined by{n� }= {u}, {n}= {u}, [M ]= [GM]− q(b/Ua)2[A3],

[C ]= [GC]− q(b/Ua)[A2], [K ]= [GK]− q[A�1], [A m ]= qpm [Am ],

[A�1]= [A1]+ sM+3

m=4

[Am ]. (17)

By rearranging the above formulae, equation (14) can be written as

[M ]{n}=−[C ]{n� }−[K ]{u� }− {f}T{ f (a)}+ sM+3

m=4

[A m ]{z� m}, (18)

z�m (t)= u� (t)− pmz� m (t), z� m (0)=0. (19)

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Combining equations (17) and (18), yields the final state space equation as

{n� ˙} −[M ]−1[C ] −[M ]−1[K ] −[M ]−1[A 4] · · · −[M ]−1[A M+3]

{u� ˙} [I] [0] [0] · · · [0]

{z� 4} = [0] [I] −p4[I] [0] [0]gG

G

G

G

F

f

hG

G

G

G

J

j

GG

G

G

G

K

k

GG

G

G

G

L

l

* * * [0]···

*

{z� M+3} [0] [I] [0] · · · −pM+3[I]

{n� } {f}T{ f (a)}{u� } {0}

× {z� 4} −[M ]−1 {0} (20)gG

G

G

G

F

f

hG

G

G

G

J

j

gG

G

G

G

F

f

hG

G

G

G

J

j

.

* *{z� M+3} {0}

Figure 7. The limit cycle flutter characteristics of flexible typical section model with a freeplay(Ka =800Nm/rad, va/vh =0·815): (a) flutter speed; (b) flutter frequency. –––, D.F. (rigid); ——, time simulation(flexible); — . —, flexible linear flutter; — . . —, rigid linear flutter.

.-. . 832

Figure 8. A comparison of time responses of free oscillation between the fictitious mass model and the fullmodel: (a) linear case (s=0); (b) non-linear case (a0/s=10). ——, Full simulation; w, fictitious mass model.

The number of equations in the matrix equation (20) is (2+M)×N (M=number ofsimple poles, N=number of normal modes). Integrating equation (20) gives the timeresponse of a non-linear system. Here, the 5–6th order Runge–Kutta–Verner algorithmis used for the adaptive integration step.

Integration of equation (20) requires an initial condition. In non-linear analysis, theresponse is dependent on the initial conditions, so the condition must be chosen carefully.Since the computation is performed in the modal co-ordinate of the fictitious mass model,the initial condition is to be expressed in terms of the modal co-ordinates. In this study,a rigid body rotation of freeplay angle and the first mode of the linear model are combinedto assign an initial condition. Therefore the initial pitch amplitude is defined as

a0 = s+ ae1, (21)

where s is the freeplay angle, and ae1 is the pitch angle produced by the deformationproportional to the first linear vibration mode.

3. RESULTS AND DISCUSSION

3.1.

A linear flutter analysis has been performed prior to a non-linear analysis to determinethe aeroelastic characteristics of the linear mode. The model consists of a uniform

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aluminum beam, a torsional pitch spring and a plunge spring, as shown in Figure 2, wherea and h are defined as the pitch angle and the plunge displacement at the elastic axis(pitching axis), respectively. The elastic axis is located at 37·5% chord. The value of theplunge spring stiffness Kh is 500 000 N/m. The deformation of the beam is assumed to besmall so that the rotatory inertia effect is neglected. The chord length of the beam 15·6 cmand the thickness of the beam is 2 mm unless otherwise specified. Eight beam finiteelements were used for structural analysis. Also, a reduced modulus of elasticity(E'=E/(1− n2)) was used for the plane strain condition. Structural damping coefficientsin all modes are assumed to be g=2z=0·01.

Linear flutter results have been obtained for both flexible and rigid models to investigatethe flexibility effect. Also, the computation has been performed for a torsional spring withvarious stiffnesses and for a beam with various thicknesses. A rectangular wing (aspectratio=20), which consists of 200 DLM elements (ten chord-wise and 20 span-wise), wasused to determine the two-dimensional aerodynamic pressure. In this analysis, a normalmode approach was used to save computation time. Four vibration modes proved to besufficient to obtain a converged flutter speed in this analysis.

In Figure 3 are shown the natural frequencies of the flexible and the rigid models, whereva and vh are the uncoupled natural frequencies of pitch and plunge, respectively, and vb

is the natural frequency for the chordwise bending mode. The natural frequencies areexpressed as follows:

va =zKa /Ia , vh =zKh /m , vb =(1·506p)2zE'I/mL3. (22)

Here Ia is the rotational inertia of the beam about the elastic axis, m is the mass of thebeam, and vb is obtained from the free–free beam boundary condition. vn is the naturalfrequency of the typical section model. In Figure 3, ‘‘rigid’’ means that the chordwise

Figure 9. The parameters map of a rigid and flexible typical section with a freeplay for air speed versus theinitial condition ratio (Ka =2400Nm/rad): (a) rigid model; (b) flexible model. F, divergent flutter; C, chaoticmotion; L, limit cycle oscillation; P, periodic motion; ,, damped stable motion.

.-. . 834

Figure 10. The parameters map of rigid and flexible typical section with a freeplay for air speed versus theinitial condition ratio (Ka =800Nm/rad): (a) rigid model; (b) flexible model. F, divergent flutter; C, chaoticmotion; L, limit cycle oscillation; P, periodic motion; ,, damped stable motion.

bending stiffness of the airfoil is infinite and the chordwise bending deflection is zero, and‘‘flexible’’ means that the airfoil is flexible. The effect of the frequency ration (va /vh ) onthe natural frequency of the typical section is shown in Figure 3(a). Here, the variousfrequency ratios have been determined by changing the torsional spring stiffness from 200to 4000 Nm/rad. As the frequency ratio increases, the flexibility effect on the first andsecond mode increases. The effect of the change in beam thickness on the naturalfrequencies of the typical section is shown in Figure 3(b). The increase in beam thicknessresults in increases in the mass and bending stiffness. The flexibility effect of the typicalsection on the natural frequency increases as the thickness decreases.

Flutter analysis was performed by using the V-g method, for sea level conditions. Thedivergence speed, the flutter speed and the frequency of the model of Figure 3(a) are shownin Figure 4. For the rigid model, the present results are very close to those ofMSC/NASTRAN, in which strip theory is used with the Theodorsen function. Thedivergence speed is lower than the flutter speed for va /vh Q 0·7. The chordwise bendingeffect on the flutter becomes significant for va /vh q 1·0. The flutter speed of the flexibleairfoil is lower than that of the rigid airfoil for va /vh q 1·0. The divergence speed is lowerthan that of the rigid airfoil. Also, the flutter frequency of the flexible airfoil is lower thanthat of the rigid airfoil.

Thickness effects on the flutter characteristics are shown in Figure 5. As the thicknessof the airfoil decreases, the flutter speeds of the rigid and flexible airfoils decrease. The

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flutter speed of the flexible airfoil is lower than that of the rigid airfoil. The flexibility effectof the airfoil on the flutter becomes significant as the thickness decreases.

3.2. -

Non-linear aeroelastic analysis was performed for the model with a pitch freeplaynon-linearity. Rigid and flexible models with the frequency ratio (va /vh ) smaller than 1and larger than 1 were selected for the computational model. The plunge spring stiffness(Kh ) was set to 500 000 Nm/rad and the pitch stiffness (Ka ) was selected to be 800 Nm/rad(va /vh =0·815) and 2400 Nm/rad (va /vh =1·412).

The frequency domain analysis using the describing function was performed first. Theeffects of the limit cycle amplitude ratio (A/s) on the flutter speed were examined and areshown in Figures 6 and 7. The case of Ka =2400 Nm/rad is shown in Figure 6. The overalltrend is similar to that of Figure 4, since the equivalent linearized pitch stiffness increasesas the amplitude ratio increases as shown in Figure 1. The minimum flutter speed isobserved near A/s=2·5. For a velocity above 150 m/s there are three types of motion:

Figure 11. The divergent flutter speed of a rigid and flexible typical section with a freeplay for air speed versusthe initial condition ratio: (a) Ka =2400Nm/rad; (b) Ka =800Nm/rad. w, rigid divergent flutter; r, flexibledivergent flutter; –––, rigid linear flutter; ——, flexible linear flutter.

.-. . 836

Figure 12. The bifurcation diagram of the pitch amplitude (Ka =2400Nm/rad): (a) rigid model (a0/s=1·0);(b) flexible model (a0/s=2·0).

static equilibrium, stable and unstable limit cycle oscillations. Stable limit cycle oscillationoccurs below the linear flutter boundary, and approaches linear flutter speed as theamplitude ratio increases. The time domain analysis gives either stable limit cycleoscillation or static equilibrium position according to the initial condition. For thestable limit cycle oscillation the results of the time domain agree well with those of thefrequency domain analysis. The frequency of the limit cycle oscillation increases as theamplitude ratio increases. The flexibility effect lowers the limit cycle flutter speed andfrequency. The results for the case of Ka =800 Nm/rad are shown in Figure 7. In this case,unstable limit cycle oscillation occurs above the linear flutter boundary. The frequency ofthe unstable limit cycle oscillation increases as the amplitude ratio increases. The timedomain analysis shows more complicated responses like chaotic motion, as will be shownlater in this paper. The simple describing function method is not appropriate for this case.

To verify the fictitious mass modal approach in the structures with the freeplaynon-linearity, the free vibration analysis for the non-linear model was performed in thetime domain. Leading edge displacements obtained by the fictitious mass model arecompared with those given by a full d.o.f. model in Figure 8. Good agreement betweenthe two models is observed. Therefore the fictitious mass modal approach is appropriatefor the modal transformation of a structure with concentrated non-linearity.

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The time domain analysis was performed for the initial amplitude ratio (a0/s) and thefree stream velocity. The results are represented in the parameter map as shown inFigures 9 and 10. Time responses are classified into five categories: damped stable motion,limit cycle oscillation, complicated periodic motion, chaotic motion and divergent flutter.Here the periodic motion means the motion of period-n or the quasi-periodic motion,including the period-1 motion which is not symmetric about the mean position. Thechaotic motion was recognized by phase plane diagrams, FFT analysis, Poincare maps,etc. Transient chaos is not considered here. The case of Ka =2400 Nm/rad is shown inFigure 9. A broad velocity region of the limit cycle oscillation is observed below thedivergent flutter speed in contrast to the linear case. The characteristics of the responsesare less affected by the initial pitch amplitude ratio. However, chaotic motion and periodicmotion appear for small amplitude ratios of the flexible model. The case ofKa =800 Nm/rad is shown in Figure 10. In this case various responses appear as the initialpitch amplitude ratio and velocity vary. The rigid and flexible models show similar trends.

The effect of the initial pitch amplitude ratio on the divergent flutter speed is shown inFigure 11. The case of Ka =2400 Nm/rad is shown in Figure 11(a). The divergent flutterspeed is almost the same as the linear flutter boundary. The effect of initial pitch amplituderatio is not significant in the rigid model. When the initial amplitude ratio of the flexible

Figure 13. The bifurcation diagram of the pitch amplitude (Ka =800Nm/rad, a0/s=1·0): (a) rigid model;(b) flexible model.

.-. . 838

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- 839

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.-. . 840

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.-. . 842

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- 843

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.-. . 844

model is small, the divergent flutter speed is higher than the linear flutter boundary. Thecase of Ka =800 Nm/rad is shown in Figure 11(b). In this case the divergent flutter speedis higher than the linear flutter boundary, and decreases as the initial pitch amplitude ratioincreases. It is observed that the flexibility effect generally lowers the divergent flutterspeed.

The bifurcation diagrams for the pitch angle of the elastic axis with velocity change areshown in Figures 12 and 13. The displacement of the pitch angle the velocity of which iszero are dotted when the motion is settled to steady motion. The case of Ka =2400 Nm/radis shown in Figure 12. Both the rigid and flexible model show similar trends. Limit cycleoscillations are observed after the damped stable motion. The case of Ka =800 Nm/radis shown in Figure 13. More complicated structures are observed than in the previous case.Hopf bifurcation and period doubling phenomena are observed. A chaotic motion bandis seen between the periodic motion bands.

The detailed time history and phase plane diagrams for the pitch and plunge motionsof the elastic axis are shown in Figures 14–19. The phase plane diagrams were drawn afterthe motion settled to steady motion. The limit cycle oscillation of the case ofKa =2400 Nm/rad is shown in Figures 14 and 15. Figure 14 is for the rigid model andFigure 15 is for the flexible model. In the flexible model, fluctuation of the pitch velocityis observed due to the flexibility effect, whereas the plunge motion shows a more smoothphase plane curve. The results for the case of Ka =800 Nm/rad are shown inFigures 16–19. Figures 16 and 17 are for the rigid model and Figures 18 and 19 are forthe flexible model. Figure 16 represents the periodic motion, and Figure 17 shows thechaotic motion of the rigid model. The shape of the phase plane diagram resembles thatof buckled plate flutter [21]. The periodic and chaotic motions are shown in Figures 18and 19. The shapes of the phase planes are similar to those of the rigid model: however,fluctuations of the pitch motion curve in the phase plane diagram are observed due to theflexibility effect.

4. CONCLUSIONS

Aeroelastic analysis for a flexible two-dimensional airfoil with a freeplay non-linearityhas been performed. Both frequency domain and time domain analysis were performedfor the rigid and flexible models. Concluding remarks from the analysis are as follows.

1. The flexibility effect in the chordwise bending deflection on the linear flutter becomessignificant when the frequency ratio of the pitch to the plunge (va /vh ) is greater than 1.The thickness effect on the linear flutter becomes significant as the thickness of the airfoildecreases.

2. From the comparison with the full d.o.f. model the fictitious mass modal approachis shown to be an appropriate modal method for the non-linear aeroelastic analysisincluding concentrated non-linearity.

3. Complex periodic motion and chaotic motion are observed, and occur more easilywhen the frequency ratio is smaller than 1.

4. The flexible model behaviour shows trends similar to that of the rigid model.However, the flexibility makes the pitch velocity fluctuate.

5. When the frequency ratio is larger than 1, the divergent flutter speed is almost thesame as the linear flutter boundary. However, the flexibility increases the divergent flutterspeed for small initial pitch amplitude ratio. When the frequency ratio is smaller than 1,the divergent flutter speed is larger than the linear flutter boundary, and approaches it asthe initial pitch amplitude ratio increases.

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REFERENCES

1. D. S. W, H. W. R and R. E. A 1957 Journal of Aeronautical Sciences24, 57–63. An investigation of effects of certain type of structural nonlinearities on wing andcontrol surface flutter.

2. S. F. S 1959 Journal of Aerospace Science 28, 25–32, 45. An approximate analysis ofnonlinear flutter problems.

3. S. C. MI, J., R. E. R, J and W. P. R 1981 Journal of Aircraft 18, 1057–1063.Experimental and theoretical study of nonlinear flutter.

4. C. L. L 1986 American Institute of Aeronautics and Astronautics Journal 24, 833–840.An iterative procedure for nonlinear flutter analysis.

5. Z. C. Y and L. C. Z 1990 Journal of Sound and Vibration 123, 1–13. Analysis of limitcycle flutter of an airfoil in incompressible flow.

6. L. C. Z and Z. C. Y 1990 Journal of Sound and Vibration 138, 245–254. Chaotic motionsof an airfoil with non-linear stiffness in incompressible flow.

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13. J. D. C. C 1974 Noise, Shock, and Vibration Conference, Monash University, Melbourne,Australia. On the hydrodynamic flutter anomaly.

14. E. A and W. P. R 1969 American Institute of Aeronautics and Astronautics Journal7, 279–285. A doublet-lattice method for calculating lift distributions on oscillating surfaces insubsonic flows.

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16. M. K and C. D. W 1994 Journal of Aircraft 31, 404–410. Time simulation of flutterwith large stiffness changes.

17. R. M. L and R. M. T 1980 American Institute of Aeronautics and AstronauticsJournal 18, 1245–1251. Flutter analysis of missile control surface containing structuralnonlinearities.

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19. I. A 1979 NASA TP 1367. An analytical technique for predicting the characteristics of aflexible wing equipped with an active flutter suppression system and comparison with windtunnel data.

20. J. A. N and R. M 1969 Computer Journal 7, 279–285. A simplex method for functionminimization.

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APPENDIX: LIST OF SYMBOLS

[C] structural damping matrixE Young’s modulusE' reduced modulus of elasticityf (a) non-linear restoring force{F} external force vector[GM] generalized mass matrix[GK] generalized stiffness matrixg modal structural damping coefficient

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h plunge displacement of elastic axisk =vb/Ua, reduced frequencyKa pitch spring stiffness[K] stiffness matrix[DK] change in stiffness matrix[M] mass matrix[Mf ] fictitious mass matrixq dynamic pressure[Q] aerodynamic influence coefficient matrix{R} elastic restoring forces freeplay angleUa free stream velocityz� m augmented statesa pitch rotation angler density[ff ] mode vector of the fictitious mass model[fb ] mode vector of the basic system[xb ] transformation matrix of the basic systemv natural frequency in radiand describing functionz =g/2, modal structural damping coefficientn Poisson ratio

Subscripts0 initial conditionb basic systemf fictitious mass model

Miscellaneous(·) =d()/ dt– (underline) generalized co-ordinate– (overbar) amplitude of harmonic motion