abstract - witpress.com flutter speed with cantilever end conditions of the wing is calculated...

Flutter sensitivity studies of high aspect

ratio aircraft wings

J.R. Banerjee

Department of Mechanical Engineering and

Aeronautics, City University, Northampton

Square, London, UK

ABSTRACT

Flutter sensitivities of high aspect ratio aircraft

wings with cantilever end conditions are studied.

The flutter speed is calculated applying a normal

mode approach through the use of generalised

coordinates. Significant structural parameters

which affect the flutter speed, are varied when

investigating the flutter sensitivities. Results

are presented for four representative aircraft

wings. The significances of results are discussed.

Transactions on the Built Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

374 Optimization of Structural Systems

INTRODUCTION

Sensitivity analyses play an important role in the

solution of optimisation problems [1-4]. In the

case of aeroelastic optimisation of an aircraft

wing, one of the aims is to achieve a lighter wing

with a maximum possible flutter speed. The present

paper addresses this problem and provides flutter

sensitivities of high aspect ratio straight

(unswept) cantilever wings when significant

structural paprameters are varied. The

investigation is focused on four representative

aircraft wings. The flutter speed with cantilever

end conditions of the wing is calculated applying a

normal mode approach through the use of generalised

coordinates [5-8]. The wing is idealised both

structurally and aerodynamically. The structural

idealisation includes beam and lumped mass

representation of the wing whereas strip theoory

based on Theodorsen type unsteady aerodynamics [9]

is used in the aerodynamic idealisation. The author

implemented the above method in a computer program

called CALFUN [10,11] which has now been used in

obtaining the results of this paper. Significant

wing properties which affect the flutter speed and

hence the flutter sensitivities are varied in the

data and the subsequent effects are examined. These

properties are : (i) bending rigidity (El), (ii)

torsional rigidity (GJ), (iii) mass per unit length

(pA) and (iv) mass moment of inertia per unit length

(pi ) of the wing. In the four case studies whichp

follow, the existing design values of the above

parameters are used first, to establish the flutter

speed using CALFUN [10,11]. Then the value of each

of the above parameters is varied independently

(increased or decreased by 25% from the existing


Optimization of Structural Systems 375

value). The subsequent effect of the variation on

the flutter speed is studied and the sensitivity

curves are drawn. The theory used in CALFUN is

briefly summarised as follows.

FLUTTER ANALYSIS USING CALFUN

CALFUN is a computer program in FORTRAN (with about

5000 lines of instructions) which uses the normal

mode method and generalised coordinates to compute

the flutter speed of an aircraft wing from its basic

structural and aerodynamic data (see Refs. [10,11]).

The normal mode method of flutter analysis used in

CALFUN is well established and has been reported in

a number of papers [5-8]. Basically the method

relies on the fact that the mass, stiffness and

aerodynamic properties of an aircraft wing can be

expressed in terms of the generalised coordinates.

Thus following the usual finite element method

CALFUN calculates the natural frequencies and normal

modes of an aircraft wing and then obtains its

generalised mass, stiffness and aerodynamic

matrices, respectively. The flutter matrix is then

formed by algebraically summing the generalised

mass, stiffness and aerodynamic matrices.

In the structural idealisation of the wing, beam and

lumped mass elements are used in CALFUN to obtain

the mass matrix [ M ] and the stiffness matrix [ K ]

of the wing. The natural frequencies w^ and the

normal mode shapes <p^ (where i is the order of the

natural frequency /normal mode) are then calculated

following the usual eigensolution procedure [7,8].



Evaluation of Generalised Mass and Stiffness Matrices

The mass and stiffness matrices of the wing are

reduced to diagonal form to give generalised mass

and stiffness matrices. This is done by using the

normal (i.e. orthogonal) modes obtained from the

finite element analysis. The procedure is briefly

explained as follows.

If [ $ ] is the modal matrix i.e. the matrix formed

by the selected normal mode shapes so that each

column of [ $ ] represents a normal mode shape <^ ,

then the generalised mass and stiffness matrices are

respectively, obtained by post multiplying the mass

and stiffness matrices by the modal matrix [ $ ] ,

and premultiplying the resultant matrix by the

transpose of the modal matrix (i.e. [ $ ] ). In

matrix notation

1 • n - n • ]

r*j- [ • n * n • ]

where [*- M __J and [^ K JL are respectively, the

generalised mass and stiffness matrices of the wing.

Clearly, if the number of modes chosen in the

analysis is n , the order of f* M J and 0 K J will

each be n x n .

Evaluation of Generalised Aerodynamic Matrix

The generalised aerodynamic matrix is formed by

applying the principle of virtual work. The

aerodynamic strip theory based on Theodorsen

expressions for unsteady lift and moment [9, 12, 13]



and the normal modes obtained from the finite

element method are used when applying the principle

of virtual work. The displacements considered are

the vertical deflection (bending) h(y), and the

pitching rotation (torsion) <%(y), of the elastic

axis of the wing at a spanwise distance y from the

root. Thus the displacement components of the ith

mode <p. are respectively, h.(y) and ou(y). If q^(t)

( i =1, 2, . ...n) are the generalised coordinates,

h(y) and cx(y) can be expressed as

MY) = Y. h. (y) q. (t)i=l * *

(3)

«(Y) = E oc. (y) q. (t)* i

(4)

Equations (3) and (4) can be written in matrix form

as

h(y)

(5)

If L(y) and M(y) are respectively, the unsteady lift

and moment at a spanwise distance y from the root,

the virtual work done (<5W) by the aerodynamic forces

is given by

n s<5W = I Sq. S { L(y)h. (y) + M(y)a.(y) } dy (6)

i=l ^ 0 i i

where s is the semi-span (i.e. length) of the wing

and n is the number of normal modes considered in

the analysis.



Equation (6) can be written in matrix form as

1

2̂

S= JT

0

%2 *2

h (%__ n n _.

" L(y)l

_M(y)J(7)

The unsteady lift L(y) and moment M(y) in two

dimensional flow given by Theodorsen [9, 12, 13] can

be expressed as

L(y)

M(y)

where

11

21

12

22 J

h(y)

(8)

-k+2C(k)ik

r (9)

(a^-l/2)ik ]

In equations (9) U, b, p, k, C(k) and a^ are in the

usual notation : the airspeed, semi-chord, density

of air, reduced frequency parameter, Theodorsen

function and elastic axis location from mid-chord,

respectively (see Refs. [12, 13]).



Substituting equation (8) into equation (7) and

using equation (5)

aw

SW

sJ0

dy

QF11_ QF12 nn

(10)

where [QF] is the generalised aerodynamic matrix

with

srv0

(11)

The generalised aerodynamic matrix [QF] is usually a

complex matrix with each element having a real part

and an imaginary part. This is as a consequence of

the terms A^, A^- •• etc. in equation (11) being

complex (see equations (9)). In contrast, the

generalised mass and stiffness matrices are both

real (and diagonal) matrices.


380 Optimization ot Structural Systems

Formation and Solution of the Flutter Determinant

The flutter determinant is the determinant formed

from the flutter matrix, and the flutter matrix is

formed by algebraically summing the generalised

mass, stiffness and aerodynamic matrices. Thus for

a system without structural damping (structural

damping has generally a small effect on the

oscillatory motion and is not considered here) the

flutter matrix [ QA ] can be formed as

where w is the circular frequency in rad/s of the

oscillatory harmonic motion.

For the flutter condition to occur, the determinant

of the complex flutter matrix must be zero so that

from equation (12)

M J + f K J - [ = 0 (13)

The solution of the above flutter determinant is a

complex eigenvalue problem because the determinant

is primarily a complex function of two unknown

variables, the airspeed (U) and the frequency (w).

The method used in CALFUN [10, 11] selects an

airspeed and evaluates the real and imaginary parts

of flutter determinant for a range of frequencies.

The process is repeated for a range of airspeeds

until both the real and imaginary part of the

flutter determinant (and hence the whole flutter

determinant) vanish completely.



DISCUSSION OF RESULTS

Four representative aircraft wings are used as

illustrative examples in obtaining the results. All

the four wings are straight and unswept and the

details are as follows. The data for the Loring

wing [6] and the Goland wing [14, 15] are taken from

the published literature whereas the data for the

other two wings are those of two existing sailplanes

namely, the Kestrel-22m and the Vega-15m,

respectively (the Kestrel-22m and the Vega-15m are

hereafter referred to as simply the Kestrel and the

Vega, although there are other versions of the

sailplanes). The aspect ratios of the Loring,

Goland, Kestrel and Vega wings are 6.75, 6.67, 31.4

and 22.4 respectively. It may be noted that both

the Loring wing [6] and the Goland wing [14] are

uniform and so, have constant structural and

aerodynamic properties along the span whereas both

the Kestrel and the Vega wings are nonuniform and

tapered and so, have variable properties along the

span. The flutter speeds and flutter frequencies of

the four wings with cantilever end conditions were

calculated using the existing design values of wing

properties. The results obtained from CALFUN are

shown in Table 1. The calculated flutter speeds for

the Loring and the Goland wing agreed completely

with the published results [6, 14, 15]. The flutter

speeds of the Kestrel and the Vega wings shown in

Table 1 agreed very well with pilot reports and

flight test results. The results are in complete

agreement with those given in the "Jane's All the

World's Aircraft" which quotes the flutter speeds of

the Kestrel and the Vega to be 71 m/s and 90.5 m/s

respectively.



Next the effects of various structural parameters on

flutter speed were investigated. Four structural

parameters were varied independently (i.e. one

parameter at a time is changed while keeping the

others constant) to calculate the flutter

sensitivities. These parameters have already been

described earlier. Each of these parameters was

varied between + 25% of the design values and the

flutter speed was calculated for each case. The

percentage variations in flutter speed against the

percentage variations of the structural parameters

are shown in Fig. 1 for each of the four wings

analysed.

Results shown in Fig. l(i) clearly indicate that the

variation in bending rigidity (El) has virtually no

effect on the flutter speeds of the Kestrel and the

Vega wings whereas it it has some small effect on

the flutter speed of Goland wing [14, 15] and a

relatively much smaller effect on that of the Loring

wing [6]. However the effect of the variation of

torsional rigidity (GJ) on the flutter speed is much

more pronounced as shown in Fig. l(ii). This is

expected because the torsional rigidity of a wing

generally plays a bigger role than its bending

rigidity in a flutter situation. Interestingly, the

slope of the graphs for all the four wings are very

nearly equal (see Fig. l(ii)) and show a linear

trend. It should be noted that an increase in the

bending rigidity (El) reduces the flutter speed

whereas an increase in the torsional rigidity (GJ)

increases the flutter speed. The probable reasons

for this are as follows. The fundamental torsional

natural frequencies of all the four wings are higher

than those of the fundamental bending natural



frequencies. Thus an increase in the bending

rigidity increases the bending frequency which in

turn reduces the separation between the fundamental

bending and torsional natural frequencies of the

wing and so causing flutter at a lower speed. In

contrast, an increase in the torsional rigidity

increases the torsional frequency and hence widens

the separation between the two frequencies resulting

in a higher flutter speed. The effects of the mass

per unit length (pA) and the mass moment of inertia

(pi ) per unit length on the flutter speed, are

found to be small, except for the Goland wing (see

Fig. l(iii) and Fig. l(iv)) for which a maximum of

10% variation in flutter speed is achievable with

the ranges of interest. It is very significant that

the flutter sensitivities shown in Fig. 1 are all

well behaved and the results indicate that further

studies on aeroelastic optimisation can be carried

out with ease and in a relatively straight forward

and uncomplicated manner, e.g. the sensitivity

information obtained in this paper can be directly

used in an optimisation program such as CONMIN [16].

CONCLUSIONS

Flutter speeds and flutter sensitivities have been

investigated for four representative high aspect

ratio aircraft wings. The effects of significant

structural parameters on the flutter speed are

illustrated by numerical results. The investigation

gives cautious optimism for optimisation studies.



Table 1. Flutter speed and flutter frequency of

high aspect ratio aircraft wing

Aircraf

(case

Loring

Golland

Kestrel

Vega Wi

t Wing

study)

Wing

Wing

Wing

ng

Flut

m

90

136

69

89

ter speed

/s

.5

.9

.6

.8

Flutter

rad

57.

70.

53.

71.

/

0

0

5

9

frequency

s

REFERENCES

1. Lu, Y. and Murthy, V.R. ' Stability Sensitivity

Studies for Synthesis of Aeroelastic Systems'

Journal of Aircraft, Vol. 27, pp. 849-850, 1990.

2. Dodd, A.J., Kadrinka, K.E. and Loikkanen, M.J.

' Aeroelastic Design Optimization Program' Journal

Of Aircraft, Vol. 27, pp. 1028-1035, 1990.

3. Adelman, H.M. and Haftka, R.T. ' Sensitivity

Analysis of Discrete Structural Systems ' AIAA

Journal, Vol. 24, pp. 823-832, 1986.

4. Bindolino, P. and Mantegazza, P. ' Aeroelastic

Derivatives as a Sensitivity Analysis of Nonlinear

Equations ' AIAA Journal, Vol. 25, pp. 1145-1146, 1987,



5. Loring, S.J. ' General Approach to the Flutter

Problem ' SAE Journal (Transaction), Vol. 49,

pp. 345-356, 1941.

6. Loring, S.J. ' Use of Generalised Coordinates in

Flutter Analysis ' SAE Journal (Transaction), Vol.

52, pp. 113-132, 1944.

7. Banerjee, J.R. ' Flutter Characteristics of High

Aspect Ratio Tailless Aircraft • Journal of

Aircraft, Vol. 21, pp. 733-736, 1984.

8. Banerjee, J.R. ' Flutter Modes of High Aspect

Ratio Tailless Aircraft ' Journal of Aircraft,

Vol. 25, pp. 473-476, 1988.

9. Theodorsen, T. ' General Theory of Instability

and Mechanisms of Flutter ' NACA Tech. Report. 496,

1934.

10. Banerjee, J.R. ' Use and Capability of CALFUN-

a Program for calculation of Flutter Speed Using

Normal Modes ' Proceedings of the International AMSE

Conference on Modelling and Simulation, Vol. 3.1,

Athens, June 27-29, pp. 121-131, 1984.

11. Banerjee, J.R. ' User's Guide to the Computer

Program CALFUN (CALculation of Flutter Speed Using

Normal Modes) ' Department of Mechanical Engineering

and Aeronautics, MEAD/AERO Report No. 164, City

University, London, 1989.

12. Bisplinghoff, B.L., Ashley, H. and Halfman,

R.L. Aeroelasticity, Addison-Wesley, Reading,

Massachusetts, 1955.


386 Optimization ot" Structural Systems

13. Fung, Y.C. 4n Introduction to the Theory of

Aeroelasticity, John Wiley and Sons, New York, 1955.

14. Goland, M. ' The Flutter of a Uniform

Cantilever Wing ' Journal of Applied Mechanics,

Vol. 12, pp. A197-A208, 1945.

15. Goland, M. ' The Flutter of a Uniform Wing with

Tip Weights ' Journal of Applied Mechanics, Vol. 15,

pp. 13-20, 1948.

16. Vanderplaats, G.N. ' CONMIN : A Fortran Program

for Constrained Function Minimization, User's

Manual', NASA TMX 62282, August 1973.


5%

-5%

Optimization of Structural Systems

T 1 T

(i)

10%

-10%

-20%

387

(ii)

-25% 0 25%% increase in El—»—

-25% 0 25%% increase in GJ ——

10%

-105

(iii)

I I

10%

-10%

J_-25% 0 25%

% increase in pA—*-

-25% 0 25%

% increase in pip—•—

Figure 1 Variation of flutter speed against(i) bending rigidity (El), (ii) torsional rigidity (GJ),(iii) mass per unit length ( pA) and (iv) mass moment of

inertia per unit length ( pip)

Key: Goland wing, - -

-Kestrel wing,

- - Loring wing,

... Vega wing.


abstract - witpress.com flutter speed with cantilever end conditions of the wing is calculated...

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