Flutter Analysis of Cantilever Composite Plates in Subsonic

7/27/2019 Flutter Analysis of Cantilever Composite Plates in Subsonic

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1102 AIAA JOURNAL VOL. 27,NO. 8

Flutter Analysis of Cantilever Composite Platesin Subsonic Flow

Kuo-Juin Lin, * Pong- Jeu Lu,? and Jiann-Quo Tarn$National ChengKung University, Tainan, Taiwan, Republic of China

The high-precision 18-degree-of-freedom triangular plate-bending finite element is used to study the effects of

composite filament angle, orthotropic modulus ratio, sweep angle, and aspect ratio on the vibration and

flutter/divergence characteristics of cantilever plates in subsonic flow. The element stiffness and mass matricesare generated according to the classical lamination theory. Unsteady airload acting on the plate is evaluated bythe use of lifting surface theory, which is solved numerically by the doublet-lattice method. T o facilitate flutter

analysis, interpolation using a surface spline is employed to interconnect the structural nodal and aerodynamic

control points. The flutterldivergence tradeoff of aeroelastic tailoring is found, in a more involved manner, forthe present plate-like low-aspect-ratio wings as compared to that associated with the beam model first discovered

by Weisshaar. Effective enhancement on flutter/divergence performance can be attained by varying the or-thotropic modulus ratio while as appropriate fiber orientation is selected. Also, as discovered previously, the

present numerical results confirm tha t the structural tailoring can provide a harmonious balance to the sweepangle effect upon the aeroelastic stability characteristics of a wing. From the numerical examinations performed,

it is concluded that substantial improvement upon flutter/divergence characteristics can be achieved by usingcomposite materials. Nevertheless, the benefit can be gained only through a thorough parametric investigation

because the aeroelastic stabilities are also complicated by the directional stiffness of composites, in part icular for

the cases of low-aspect-ratio wings.

I. Introduction

LUTTER and divergence phenomena are defined respec-Ftively as the dynamic or static instability of an aeroelasticsystem, characterized by the interactions of elastic deforma-tion and aerodynamic loads. They are aeronautically veryimportant since the onset of flutter/divergence may quicklydevelop into catastrophic structural failure or undesirablelimit-cycle oscillations. In addit ion, the modern aircraft designrequiring high speed with lightweight structure makes theaeroelastic analysis a stringent necessity in the design stage.Employing composite material in the manufacture of airc raftstructure has been one of the avenues to enlarge the flutter/divergence boundary of an a ircraft. This is due to the promi-nent characteristics associated with composite material,namely, the high strength/stiffness-to-weightratio and thecontrollable material properties that usually are not possessedby conventional materials. Studies using composite structuresto improve the aeroelastic behavior, divergence, or flutter, astermed by aeroelastic tailoring, have been conducted by manyresearchers, and all investigations confirmed the benefit of

composite material in enhancing the aeroelastic properites. l I1Reference 1 presents an excellent review of aeroelastic tailor-ing and many useful references are listed therein.

Wei~shaar*.~employed laminated box beam theory and two-dimensional, incompressible strip aerodynamic theory to pre-dict the divergence characteristics of swept wings. He con-cluded that the bending-torsion stiffness coupling of anisotropiccomposite materials could be used to alleviate divergence fora large range of forward sweep angles. This conclusion wasverified in wind-tunnel tests by both Sherrer et al.4 and Blairand Wei~shaar .~Weisshaars model was further adopted forflut ter analysis by Lehman6and Lottati. Lehman showed thata strong dependence of composite filament angle was found

Received June 18, 1987; revision received Aug. 30, 1988. Copyright@ 1988 American Institute of Aeronautics and Astronautics, Inc. Allrights reserved.

*Graduate Research Assistant, Institute of Aeronautics and Astro-nautics.tAssociate Professor, Institute of Aeronautics and Astronautics.SProfessor and Chairman, Department of Civil Engineering.

for the flutter characteristics. Lottati discovered tha t the effectof bending-torsion stiffness coupling makes a compromisebetween flutter and divergence characteristics for varioussweep angles. For plate-like wings, Rogers et al.*and Braymenet al.9performed work in the aeroelastically tailored liftingsurfaces concerned mainly with static aerodynamic improve-ments. In spite of their efforts, the amount of analytical andexperimental investigations put forward for flutter character-istics of plate-like composite wings has been very limited.Recently, a series of investigations for both the divergence andflutter behaviors of composite cantilever plates were con-ducted by Hollowell and DugundjilO and Landsberger andDugundji. The analytic investigation used a Rayleigh-Ritzformulation together with incompressible three-dimensionalWeissinger L-method aerodynamics for the divergence and anincompressible two-dimensional unsteady strip theory for theflutter. The results indicated that divergence and bending-tor-sion flutter appear at low angles of attack, whereas torsionand bending stall flutter occur at higher angles of attack.

The present work attempts to develop a more general ap-

proach for the flutter analysis of plate-like composite struc-tures (divergence can be considered as a special case of flut -ter). A high-precision triangular finite-element method is usedto discretize the structure, while bearing in mind that thefinite-element method is more versatile in treating generalcomplex geometries. Three-dimensional, unsteady compress-ible lifting-surface theory and the doublet-lattice method areadopted for the evaluation of aerodynamic forces. The em-phasis of the present study is placed on the methodologydevelopment, although a simple plate-like composite structureis assumed for the structural model. Therefore, the extensionof the present method to more practical structures can bestraightforward. The effects of composite filament angle, or-thotropic modulus ratio, aspect ratio, and wing sweep angleon the vibration and flutter characteristics of a compositecantilever plate in subsonic flow are examined numerically.The flutter/divergence tradeoff discovered in the present re-sults for low-aspect-ratio wings is found in a more involvedmanner than that found for the beam-like wings. Other effectsassociated with the composite material parameters and thewing geometric shapes were also investigated and explained. It


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AUGUST 1989 FLUTTER ANALYSIS OF CANTILEVER COMPOSITE PLATES 1103

was found that substantial improvement on the flutter/diver-gence performance of wings can be achieved if the compositeplate is properly tailored.

11. Analysis

A.

The derivation of the equations of motion for flutter analy-sis is most conveniently obtained through the use of Hamil-tons principle. Employing Hamiltons principle for a non-conservative elastic system would result in the followingvariational form that governs the dynamics of the system:

Hamiltons Principle for a Nonconservative System

6 L d t + 6 W N Cd t = 01: 1:whereL is the Lagrangian defined asL = T - U, T the totalkinetic energy of the system, U the total potential, or strainenergy of the system (the gravitational potential is neglectedfor the present case), 6WNc the virtual work done by thenonconservative force field (for this case, it is the work done

by aerodynamic forces), and tl and t2 the initial and final timeof the motion.The LagrangianL will be deduced in the next section basedon the finite-element technique, from which the system massand stiffness matrices are formulated. As for the virtual workterm, 6WNc, which involves the use of doublet-lattice methodfor the unsteady lifting surface theory, the formulation will beexplained and shown subsequently in terms of the aerody-namic influence and the t ransformation matrices in Sec. 1I.Cand D.

B. Structural Finite-Element Discretization

The high-precision 18-degree-of-freedom triangular plate-bending finite element (T18 element) is used here to generatethe mass and stiffness matrices associated with the plate-likestructure. This method, originally developed by Argyris eta1.,I2 Be11,13and Cowper et a1.,I4 involves a large matr ix inver-sion and numerical integration and hence is very time consum-ing in application, although it has the advantage being con-forming and accurate. Such a drawback in inefficiency wasremedied by Stavitsdy et al.I5via the use of an oblique coordi-nate system while analyzing creeping flow problem. Recently,Jeyachandrabose and Kirkh~pe ~ , succeeded in using thisimproved method to analyze plate problems and thus refor-mulated the associated stiffness matrix of the thin plate. It isthis improvement that makes the triangular finite element oneof the most efficient and reliable thin plate elements. Thishigh-precision triangular element has another property, beinggeometrically versatile, which is of particular value for devel-oping methods in coping with problems having complex con-

figurations. Based on these reasons, this T18 element tech-nique is employed in the present work to treat the static anddynamic aeroelastic problems of laminated anisotropic plates.

In the following derivation the x,y coordinates and all thedeformations are nondimensionalized by the half root chordb, of the plate. The transverse displacement field w, within aTI8 element, can be expressed in the formI6

where [q 1 is a column vector defined in terms of the general -ized coordinates associated with the three vertices of the trian-gular element,

where Q i ) T is composed of the displacement, slopes andcurvatures of the element at the ith vertex. The column vector

( A ) consists of complete quintic polynomial expressed in

terms of the oblique coordinates (t,r))that guarantees theelement to be conforming and is given by

( A J T = L1 4 r ) [r ) v2...t5...q5J[Q] is a 21 x 18 transformation matrix related to the elementgeometry. The explicit form for [Q] can be found in Ref. 16.

According to the classical lamination theory, the kinetic and

strain energy of the structural element made of symmetricallylaminated composites are derived as

T = hb,?psts W2 dwdy (3)iU = K D 1 1 1 ~ ~ l x l ~ [ ~ l ( x lxdy 4)

In the formulation, ps and ts are the density and thickness ofthe plate element, respectively, A the area of the element, [D]the symmetric flexural rigidity matrix of the laminated plateIsnormalized by the x-axis rigidity Dll and x)the curvaturevector given by

( x I T = Lwxx wyy 2wxyJSubstituting Eq. (2 )into Eq. 4), Jeyachandraboses16obtainedthe following form for the strain energy of the element:

The definitions of matrices [Q1]and [F,] are given in Ref. 16.For the present flutter analysis, one has to derive for the

mass matrix to account for the inertia effect. SubstitutingEq. (2) into Eq. (3) and after some manipulations, the kinetic

energy of the element may be derived in the form

T = 1/2b;psts(2A) 41TIQITIAl [ e l (QJ (6)where the 21 x 21 symmetric matrix [A] is given by

(7)Taking the variation of the time integral of the Lagrangian

L, the mass and stiffness matrices of the TI8 element canreadily be found to be

[MI, = b,? @A) Q 1 [AI [Ql (8)(9)

C. Aerodynamic Theory and the Doublet-Lattice Method (DLM)Unsteady aerodynamic model is known to be very crucial

for flutter calculations. In subsonic fluid flows, wake plays avery critical role in the determination of the aerodynamicdamping effect. It has been found that an inappropriateaerodynamic model often causes substantial error in flutteranalysis. 1 9 ~ 2 0

The present work uses the linearized potential flow modelthat, as a consequence of assuming that the structure is under-going simple harmonic motion, can be expressed by a singularintegral equation connecting the downwash and the pressure

distribution on the lifting-surface as


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1104 LIN, LU, AND TARN AIAA JOURNAL

REAL PART

+ PRESENT METHOD1 A AL BA NO & RODDEN i 241

0

-1 J0.0 0.2 ' 0.4 0.6 0.8 1 ox l c

a) At wing root

EXPT : N A S A T N - D - 3 4 4 [ 1 2 5 1REAL PART[MAG PARTPRESENT METHOD

THEORY:

A AL BA NO & R OD D EN 124A?M ~ 8 . 2 4k =0.47Y I S 10 9

.o .2 .4 6 8 1.X I C

b) Near wing tipFig. 1mode.

Lift distribution on rectangular wing oscillating in bending

wheres is the wing area, V the freestream velocity, and wuandACp the downwash and pressure coefficient differential acrossthe wing, respectively. The bar ) denotes that the variablebeneath is complex valued, carrying information of both mag-nitude and phase shift, K is the kernel function representingthe downwash at a location x,y,z) induced by a unit impulseload at the position (xl,ylrzl).The form of the kernel functionused herein is the one suggested by Landahl.21

The unsteady aerodynamic code used for the present flutterproblem was developed by Chu and Lu.22,23It is a wing-bodycombination code developed along the line of doublet-latticemethod (DLM) together with the interior body singularitymethod. Several validity tests are performed for a cantileveredwing, and the results are shown in Fig. 1. The comparisons aremade with those calculated by Albano and R ~ d d e n ~ ~and thoseobtained by e ~ p e r i m e n t . ~ ~The agreements are quite satisfac-tory in general.

Employing the DLM procedure, the nonconservative virtualwork term 6WNc contributed from all the aerodynamic ele-ments can be summed up to yield

6WNC= b,? 6WV [ aS ] (A.B] 1 1)Note that (61 ) is the displacement vector evaluated at eachcontrol point of the aerodynamic element, and the middlediagonal matrix is defined with the element area sj aligneddiagonally.

- Y

0AERODYNAMIC CONTROL POIN T

0 COMPOSITE FILAMENT ANGLEX

Fig. 2 Structural and aerodynamic networks.

D. Interpolation between Structural and Aerodynamic Networks

The discretization of structural and aerodynamic modelsusually leads to two sets of nodal and/or control points thatform two different networks. Aerodynamic panels, which al-most exclusively adopt quadrilateral shapes, mustbe orientedwith their side edges parallel to the airstream direction, owingto the specific physics inherent in discretizing the lifting sur-faces. The present work uses triangular finite elements forstructures and hence results in a different set of locations ofthe nodal points as opposed to those of the aerodynamiccontrol points. A schematic is illustrated in Fig. 2. Amongvarious methods used in interconnecting these two networks,the one developed by Harder and DesmaraisZ6proves to be themost successful. The interpolation function used by Harderand Desmarais is the surface spline function derived from an

infinite thin plate pinned at the fitting points. Physically, thisis equivalent to making the mathematical interpolation be-tween the fitting points more compatible with the actual de-flection of the structures. Therefore, the downwash at theaerodynamic control points, as derived from this interpolationfunction, can be more accurately approximated, and the spuri-ous wriggles that may result in a completely erroneous evalua-tion of the air loads acting on the structure can be avoided.

The detail derivation of the interpolation function is elabo-rated in Ref. 26. The relation between the locations of theaerodynamic control points and the structural nodal pointscan be established using some transformation matrices [TI]and [T2] ,which, as inserted into the expression of ( 6 ~ 1and(AD 1, yields the nonconservative virtual work 6WNc of theform

where pa is the air density and [A]-' the inverse of the aero-dynamic influence coefficient matrix. The detail derivationleading to Eqs. (12) can be found in Ref. 27.

E.

Assembling together the previously derived element massand stiffness matrices of the structure and the aerodynamicforce acting on it, the equations of motion can be obtained.

Furthermore, by adopting the assumption of simple harmonicoscillation, for which the onset of flutter is defined, the flutterequations can be derived as

Flutter Equations and Solution Method


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AUGUST 1989 FLUTTER ANALYSIS O F CANTILEVER COMPOSITE PLATES

1ST MX)E0 I I I I I

1107

-

2.0

FLUTTER

00 30 60 90 120 150 180

0Fig. 6a Normalized flutter and divergence speeds VF /VR andV D / V Rvs composite filament angle 0 forAR = 1.5 (j3= 4, structuralmesh = 3 X 4, aerodynamic mesh = 5 x 5, VR = 138.47 m/s = VD ate =o deg).

All = 1.5

,/---

'

-------

range into two regions bounded respectively by monotonicallyvarying stability boundaries as illustrated in Refs. 1 and 29.Instead, for the present plate-like model, the high-flutter/low-divergence and high-divergenceAow-flutter boundary zones,confined by wavy shape curves, generally appear to describethe tradeoff phenomenon. It is observed that, generally, highflutter speed occurs for 890 deg. This observation complies with the previous find-ing that the fiber orientation shows an opposite trend com-pared to the effect of the wing sweep angle. As a whole, it isconcluded that varying the directional stiffness could be eitherbeneficial or detrimental for plate-like wings, and a carefulparametric study is critical because the instability characteris-tics are complicated by the use of composite materials.

Another point worth mentioning is the asymmetry of theflutter frequency with respect to the composite orientation.The asymmetric flutter frequencies are shown in Figs. 5b and

6b where the structural natural frequencies are seen to besymmetric. This asymmetry in the flutter frequency with re-spect to the composite filament angle is attributed to the factthat the unsteady air load is not symmetric fore-and-aftbecause of the Kutta condition required a t the trailing edge ofthe wing.

s3\

1.61 Jv1.5-1 . 4 -

1.3-1.2-

1.1-1.0-0.9

10 12 14 162 4

Fig. 7 Normalized flutter speeds VF / V Rvs orthotropic modulus ra-tio p for AR = 1.5 (structural mesh = 3 X 4, aerodynamic mesh =5 X 5, VR = 108.7 m/s).

I v

0.8 / I I I I I I I10 12 14 160 2 4

Fig. 8ratio 6 forAR = 1.5 (structural mesh = 3 X 4, aerodynamic mesh =5 x 5, VR = 139.99 m/s).

Normalized divergence speeds V D / V Rvs orthotropic modulus

C .

Figures 7and 8 show the effect of orthotropic modulus ratiop on the flutter and divergence boundaries of a wing withAR = 1.5. The values associated with the vertical coordinatesare all normalized by the corresponding flutter or divergencespeed of the isotropic cases. In the case of flutter (see Fig. 7),increasing is found beneficial in enlarging the flutterboundary. Except for the case of 0 = 0 deg, an increasingvalue may result in the increase of flutter speed, and theamount of improvement depends strongly on the filamentorientation.

Figure 8 shows the case of divergence for the same wingstructure examined in Fig. 7.It is seen that the change in thedivergence speed is not monotone while enlarging the or-thotropic modulus ratio. This phenomenon is different fromwhat were experienced in previous examinations of flutter.

The above observations imply that the orthotropic modulusratio, under certain filament angle orientation, could inducecontradictory effects upon flutter and divergence perfor-mances. For example, in Fig. 9, which is a case of 8 = 30 deg,the flutter boundary increases and the divergence boundarydecreases while enlarging the p value. Therefore, the optimal/3 value here is 4.4, which corresponds to the intersection ofthe two curves. Beyond /3 = 4.4, the low-divergence boundarywill offset the gain in the flutter improvement. Therefore, asimultaneous account of the effects of the composite or-thotropic modulus ratio and filament orientation is necessaryfor an optimal tailoring of the structures, although using

Effect of Orthotropic Modulus Ratio


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1108 LIN. LU, AND TARN AIAA JOURNAL

7 -

6-s

-s94 -

ss

V

--.-2-

1 -

1.7I

1.6 1 av1.5-

.4-1.3-1.2-

ss- c .

1

1FLUTTER

~ DIVERGENCE

0.9

10 12 14 160 2 4

Fig. 9 Normalized flutter and divergence speeds V F / V Rand VD / V Rvs orthotropic modulus ratio j3 forAR = 1.5 at 8 = 30 deg (structuralmesh = 3 x 4, aerodynamic mesh = 5 X 5 , VR = 108.7 m/s).

\\

FLUTTER

DIVERGENCE--_-\

\

0 1 I I I I I0 1 2 3 4 5A R

Fig.10 Normalized flutter and divergence speeds V F / V R andV D / V Rvs aspect ratioAR for isotropic wing VR = 38.02 m/s = VDatAR = 4).

composites can provide a latitude for improving the flutterand divergence performances.

D. Effect of Aspect Ratio

In Figs. 10-12, the effects of aspect ratioAR on the flutterand divergence boundaries are illustrated for both isotropicand composite cantilevered plates. Figure 10 shows theisotropic case p = 1) where the flutter and divergence speedsare normalized by the divergence speed atAR = 4. It is seenthat both the flutter and divergence speeds increase monoton-ically as AR decreases. The increasing rates of flutter anddivergence speeds are substantial when A R is less than 1.5.

For composite wings, the percentage increments of flutterand divergence speeds over the isotropic wing are plotted inFigs. 11 and 12, respectively. It is found that the f lutterincrements due t o the use o f composites are usually significantand less sensitive with respect to the AR value (see Fig. 11).However, the divergence increments, as shown in Fig. 12, may

become negative with the increasing value of the aspect ratio.Moreover, the occurrence of the negative divergence incre-ment is seen to be pretty parametrically dependent upon boththe composite filament angle and the orthotropic modulusratio.

E. Effect of Sweep Angle

It is known that forward- or backward-swept wings exhibitdifferent types of instabilities. Making wings sweep forwardwill result in a structural wash-in, which in general lowers thedivergence boundaries. For swept-back wings, however, theflutter boundaries become lower owing to the associated struc-

ho I

50I I I1 8 = 150-I-- I I I I0 1 3 4 5H

Fig. 11 Percentage increment in flutter speeds of composite wing vsthe aspect ratioAR V F ~= flutter speeds of isotropic wing plotted inFig. 10).


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Flutter Analysis of Cantilever Composite Plates in Subsonic

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