Flutter analysis of a wing with concentrated inertias by a direct matrix method

(1)

3 juli 1915

ARCHIEF

SYOSIUM ON STRUCTURAL DYNAMICS

PAPER NO. E.5

FLUTT ANALYSIS OP A WING WITH CONCENTRATED INERTIAS BY A DIRECT TRIX METHOD.

V.T.Nagaraj and. D.J.Johns

Loughborough University of Techno1oy) SUMMARY

A Direct Matrix method is used to predIct the flutter characteristics of a uniform cantilever wing with concentrated inertias. This method makes use of the inertial, structural and aerodynamic data of the wing in terms of the matrices of influence coefficients. These are combined to yield the flutter characteristics of the wing-inertia combination as

the elgenvalues and eigenvectors of a complex non-hermitian matrix. This eigenvalue problem is solved by an iterative technique which makes allowances for the existence of close-valued or of equal roots. The flutter characteristics (flutter speed, frequency and the mode pe) are obtained by solving the characteristic equation for a number of values of the

reduced frequenôy.

This method is applied to obtain the flutter charactoris-ics of a uniform wing with concentrated inertias. Ten control points were used in the analysis and the matrix of the aero-dynaic influence coefficients was set up by using strip theory derivatives. The rcsults obtained by this method are compared with those

obtained from wind

tunnel tests and from an assumed mode method, It is found that the values predicted

by the Direct Matrix method show

good agreement with the wind

tunnel results and are

generally on the

conservative side.

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Deift

NOTATION E a

[a

_:

Flexibility Matrix See Eq.(14) See Eq.(15) E.5.1

(2)

b VYing Semichord

[ch]

Aerodynamic Influence Coefficient Matriz

Column Matrix of Forces a the Contrai Pointe Aerodynamic Component of F

1 Sse Eq.(15)

{F Inertial Component of F

{

r}8

See Eq..(14)

g Coefficient of (fictitious) Structural Damping { h Column Matrix of Control Point Deflectiona

{h\a See Eq.(15)

Seo Eq..(14)

(Moment of Inertia of Concentrated Inertia)/(Total Wing Pitching Inertia)

I Interpolation Matrix for Deflections (See Eq.16)

[i

Transformation

Matrix for Deflections (See Eq.1l) 5LZPL

Oscillatory Aerodynamic Lift and Moment Coefficients

L1Mass iatrix

Lunpea Mass

(Weight of the Concentrated Inertla)/(Total Wing eight)

o Wing Span T Kinetic Ener

LuJ

Dynamic Matrix (See Eq.6) V Flutter Speed

Width of a Spanwise Strip

z Deflection of a point on the wing > Eigenvalue ( .>.t x

Reduced Frequency (= c b/V)

Density of Air

Flutter Frequency

x = 2b = distance of e.g. of conc. inertia aft of elastic axis.

p p

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1. INTRODUCTIOIT

It is well known that the aeroelastic characteristics of a wing can be radically altered by the addition of concen-trated masses such as fuel t'ice podded engines etc. This problc oudubecone serious when the concentrated inertia assumes large v3lues. For examples in a desi study (Ref.1) it was found that by the addition of two large lift engine pods ( 1.5, I.1O.0), placed near the two-thirds span positions the flutter spee1 was reduced fron 658 knots to

164 knots.

Por wings of conventional design, it is now possible to predicts with reasonable accuracy,the values of the flutter speed and frequency. It is aleo possible to obtain fairly accurato estimates of the effects of changing certain para-motero (e.g.9 wing mass, moment of inertia, chordwise position of the contre of gravity etc.) on the wing flutter speed. This is no longer true for the case of wings with added masses,

in spite of the fact that this topic has received a great deal of attention fron various InvestIgators (e.g., Ref s. 2 to 6). The problem of formulating a set of rules for the prediction of the flutter speeds of a wing with added masses jo complicated duo to the number an range of the parameters which can be varied both independently and simultaneously.

Molnou

(RcÍ.2) has analysed some of the published data on the flutter of wings with concentrated ±nertlas and has reoommsnded a set of modes which should te included in a flutter analysis in order to obtain reasonably accurate

estimates of the flutter speed.

Runyan and Jatkins (Ref.?) have

given an 'exact' solution

to the flutter of a uñIf orn wing with an arbitrarily placed

concentrated maso.

This method gives accurate results bu

is tedious for practical applications.

The method used here is due to Rodden (Ref

3. 8 and 9).

In this method the inertial, structural and aerodynamic

properties of the wing are used In, the fprm f matrices of the respective influence coefficients in setting up the

equations

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of motion. It is not necessary to assume the mode shapes of the fluttering wing and, if needed, these can be obtained as part of the analysis.

2. THE

DIRECT

MkTRl:

METHOD

Consider

the wing

hovin in Pig.(1).

The deflectIon

inte-gral equations of motIon

can

be written in matrix

form as:

th = (i)

where the h, a and P correspond to a chosen set of control stations. The matrix P is made up of the inertial

and

the aerodyn'n1c forces:

=

+

{a

+ {Ya (2)

If we define a complex matrix of

oscillatory

aerodynamic influence coefficients Ch such

that

{a

= Ch

b

(3)

we

can

write Eq.(2), (for harmonic motion), as

=((

[u

bZ

s

(4)

Substituting Eq.(4) into Eq.(1)

and dividing

the static flexibility coefficients by (1 + 1g) to account for the

structural damping necessary to sustain the assumed harmonic motion, we get

the

matrix equation for flutter as:

[h73

=-

[a] ([M] + s

[chi)

(5)

or,

tu\

= ).. (6)

where =

(]

₊ b s

[Ch])

and

= (i +

Eq..(6) can be solved for the complex mode shape and the complex elgenvalue

(=)

+ i ).

Prom >. , we obtain the flutter frequency, c ,

and the

required structural damping, g, as

(5)

=- )'-/ (8)

The aerodynamic influence coefficIent matrix, _Ch, requires the assumption of a reduced frequency ) for its formulation.. Prom thIs, we get the flutter speed as:

V =

(.Sb/)

₍₉₎

A curve of g vs. V can be constructed from a series of solutions of Eq.(6) for different values of the reduce& frequency. The flutter speed. is obtained as the speed at which the value of g is he same as the structural damping in the system.

The formats oÍ' the structural, inertial and aerodynamic influence coefficient matrices will now be discussed.

2.1. TEE MASS M&TRIX

Eq.Ç1) is satisfied at a set of preselected contro. stations. It is desirable to have a common set of ccntrl points for the structural, inertial, and aerodynamic influence coefficients. _{However, the choice of coitrol point locations}

is usually governed by aerodynamic considerations. (Por example, specific spanwise and chordwiee locations). _The mass matrix has to be calculated from the

dynamic properties

of each control station. These aro: the total mass, e.g. position, and. the moment of inertia. It is necessary to transform these into an equIvalent system of lumped masses. At a given station, these properties can be matched by a

system of three non-c ollinear masses having arbitrary coordi-nates. However, since the location of control stations is somewhat limited by aerodynamic considerations, a better

representation would be to use six concentrated masses having fixed coordinates.

Consider the wing shown in PIg (i). At each control station, it is assumed that the inertial properties are

represented by a system of three lumped masses M,

-Ç(Pig.2).

It is desired to find a system of equivalent

(6)

energy when oscillating at the same

frequency. (The

'dynamica-lly equivalent' mass matrix).

In terms o± the masses ( and the deflectons z,

the total

kinetic energy of the wing i given by:

The d.eflections z can be expressed in terms of the

control point deflections h by a transformation matriz

as:

=

The kinetic energy becomes:

=

Ç [i

iTM*

(12)

The inertial forces are obtaIned from Eq.(2) by using Lagrange's equations:

{r

= -

₌

_{-. [] [}

(13)

where

[ri

= _1E1 IIM*J

The dynamically equivalent miss matrix M] is symmetria bist not necessarily diagonal.

In Table (i), the mass matrix

for the model wing shown

in ?ig (4) is given0

2.2 THE TRUCTUEAL INFLUENCE COEFFICIENT MATRIX

In most cases, the structural Information is availGle as a stiffness or a flexibility matrix with respect to a

certain set of points. If these do not coincide with the set of control stations used for the aerodynamic influence coefficients, a transformation matrix has to be found.

DenotIng by the subscript 'a' the aerodynamic net and

by the subscript t3 the structural net, we are given a flexi-bility matrix

_La]

such that

(14)

It is desired to find an equivalent matrix [a _a such that a = \a]

ih

a (15)

E5.6

T

_=4.

T

r*:1

_{

(1.0)

(7)

We now define a deflection interpolation matrix [Ia

such that

=

i'a

a

(16)

Then, by using the principle of contra-gradience,

we

get, from Eq.(16)

_{_t\T}

j

}a

(['a

) s

...(17 Substituting for from Eq.(14) and for{hI from Eq.(16}, we get P'j

a as

a

(E1aLa'

_a

Conparing Eq.(18) with Eq.(15), we get as

=

([It)f

['a«'

Hence, by inversion2 T [aj a = [a]

[±1

(19)

which is the required

transformation

matrix.

In Table (i), the structural influence coefficient matrix for the uniform wing shown in Pig.(4) is given.

2.3 _{THE AERODYNM!IC INFLUENCE COEEFICIE1T} _ATRI!

(16)

Por use in the Direct atrix method, the aorodynic influence coefficient matrix

_[Ch

is defined from the equation

[P ₌

ft

bs

_[òhI

¿h

(20)

The form of the aerodynamic influence coefficient

matrix

depends upon the theory being employed for

the oscillatory

aerodynamic forces. In the

following, the strip

theory formulation of the aerodyn1c influence coefficients will

be considered.

Pig.(3) shows the given system of

deformations and the forces acting at

the qiarter chord point and also the

corresponding equivalent configuration

required in

the

E.5.7

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matrix formulation. Since the lift and the moment are given as derivatives referred to the wing quarter cbord it is convenient to take this point as the forward control point location. The location of the rear contol point can be arbitrary, bitt in this case it Is taken as the three-quarter chord poInt. In terms of tabulated quantities the

occillatory lift and moment on a strip of width Ay are

are given by

01L

(21)

LM)

Lo. bJLMz

MJLb.)

The load equivalence between the force systems is given by T

}

[1 _[Fi.? I = o

btF2J

The geometric equivalence between the deflections is given by

x

=1'

0\fh,

L-'

iJ

C.2

Prom Eqs.(21), (22), and (23), the relation between the forces P and the deflections h can be obtained as

{

P

1 _1 L

L1

f1

01

fh4

Lo 1J M7

_L-'

1j h2

(24)

Comparing Eq.(24) with Bq..(20), we get the matrix of aerodynamic influence coefficients :or the strip of width

Ayas

(22 ) (23)

Ii

-il

FL

L1 Ii

o

\I.c\

h

=I

H

L0 J _L

_L-'

Por the complete

wing, the matrix of

the aerodynanie influence coefficients appears in partitioned form as;

E. 5.8

(9)

9h\. 2

[chifl

(26)

The mat'ix for the wing shown iii Pig.(4) is shown, for )= 0.1, in Tablo(i)

2.4 SOLUTION OP THE EQUATIONS OF MOTION

Once the matrices [M] , , and [Chare formulated,

they can be assembled to obtain the dynamic matrix from Eq.(6). Due to the presence of the aerodynamic matrix,Uj will bo complex and non-hermitian. There eist a number of

methods of obtaining the elgenvalues and elgenvectore of euch a matrix. In the flutter problem, wo are interested in obtaining òly a few roots, starting with the root

having the largest magnit.udé and an iteration procedure is ideally suited for this purpose.

Since [u] is a complex matrix, the iteration procedure presents certain convergence difficultIes. Even though the

(complex) elgenvalues may all pave different values for their real and imaginary parts it Is likely that the

moduli of the roots may have eq.ual or nearly equal magnitu-doe. This hae to be anticipated and allowance be made to

overcome the difficulty.

Rodden et al. (Ref.9) and Gllnitz et al. (Ref 10) have given iterative routines for the solution of a complex non-hcrmitian matrix. In both cases, the calculations aro carried out in real, arithmetic, (in sorne places in double-precision), with the rules of complex algebra being followed.

In the present case, the method of Gollnitz was for the colution of the matrix LU] . In this method, an

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initial estimate of the eigenvalues is

obtained by the Power

method and by sweeping the Imown eigenvalues out the matrix.

To improve the accuracy of these esimates

V.nielan.dt?s

reciprocal iteration is used. The convergence difficulties due to a pair of close maitude or equal roots is overcome by Bodewig's method.

Th±sprocedure gave consistent results in all the cases

investigated in connection with the title

problem.

3. APPLICATION TO A MODEL WING WITH CONCENTRATED INERTIAS

The Direct Matrix method wa.s used to obtain the flutter speeds of the model wing-inertia combination shown in Pig.(4). This wing was of sectional construction and had a single

spar at 0.35e from the leading edge. Other details of the wing are given in Fig.(4). The concentrated

inertia

was

represented by a pod which could be attached to the wing at a number of spanwise stations.

For the theoretical analysis, the inertial

properties

of the wins were lumped at five spanwiso otations (Pig.1), and, at each station, two control points were located.. The forward ones were on the quarter chord point and the rear onìes on the three-quarter chord point, giving ten ont'l points in all.

Por ease of computation, the aerodynamic influence coefficient matrix was split up into its real and imaginary parts ( Real

_Lchl

+ i Im. t0hJ

) The matrix U was set up as follows;

Real

[u]

= [a] ( {M] + e ( Real LChl

ImaginaryJ

= b s [a] (Imaginary

Lcj

)

_(2?)

The two components of the matrix

U]

were then assembled

as a (10x20) real matrix as shown below:

u.

= Real (u. .)

u

= Im. (u11)

,

i =

j = 1,2,...10

E.5.1O

(11)

The e!genvlues and eigenvectors were obtained for a number of values of the reduced frequency _{(i.> =}

bJv)

_and the flutter speed obtained from a

v-g

plot. (The value of g for flutter was taken as zero).

A number of

wing-inertia combinations were analysed

by the Direct Matrix

method for their flutter charateristics.

Of these _{two cases are shown in Fig.(5) of the variation of}

the flutter speed for a number of positions of a concentra-ted inertia along the wing span. On the same figure are also shown the values obtained from wind tuel tests. The relevant details of the two concentrated inertias are:

Fig,5.a: M = 1.0

T

= 10.0 0

Pig.5.b: M = i.0 I = 10.0 x = -0.i

In both Fig,(5.a) and Fig.(5.b), the values of the

flutter speeds obtained

by an assumed mode method are also

given. Io modes were used in all

the calculations. These

were: a)

The fundamental bendïg mode of an equivalent canti-lever beam with a concentrated flass and b) the fundamental

torsiona. mode of an equivalent shaft with a concentrated inertiaQ

4. DISCUSSIN OP THE RESULTS A1D CONCLUSIONS

The curves of J- shown in Figs.(5.a) and (5.b) show similar trends. _{As the concentrated inertia is moved} span-wise from the wing root, the flutter speed Initially decreases. After reaching a minimum value when the concentrated inertia

is approximately at

the midepan, the flutter speed increases

again as the concentraed inertia is moved

further towards

the tip.

In both cases, the Vs predicted by the Direct Liatrix method are lower than the experimental values. This was

true for all the

cases analysed. Also the flutter speeds predicted by the Direct matrix method show reasonable agree-men with the experimental values. In general, the values predicted by the energy method lie between those predicted

by the Direct Matrix method and the experimental values.

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One of the main advantages of the Direct Matrix method is that it obviates the need for assuming the mode shapes of the fluttering wing in advance0 In fact,theso can be obtained as part of the analysis if needed. Ii' the matrix

of the aerodynamic influence coefficients is et equal to zero, Eq.(6) can be used for the vIbration analysis of the cantilever wing.

From the experience gained in this series of analyses, it can be concluded that the Direct Matrix method can be used with confidence to obtain conservative estimates of the flutter speeds of wing-inertia combinations.

The formulation also allotis the use of more sophisti-cated theories for representing the oscillating aerodynamic forces.

REFERENCES

JOHNS,D.J.

'Recent Aeroelastic

Studies at the

College of Aeronautics' CoA Noto Aero.

No. 168, Jan.1965.

MOLYNEUX,W.G.

'Flutter of t7ings with Locailsed Maeses'

R.A.E. Rep. Struct. No. 214, 1956.

GAtTKROGER,D.R. 'A Theoretical Treatment of the Flutter

of a Viing with a Localised Mass' .R.A.E. Rep. Struct0 o, 213, 1958.

ILTS,C.H. 'Incompressible Plutter Characteristics of Representative Aircraft Wings' NASA Rep. No. 139O, 1957.

GAUKROGER,D.R. 'find Tunnel Tests on the effects of a Localised Mass on the Flutter of a Swept Wing with Fixed Root' A.R.C. R.and M.

No.. 3141. 96O.

RUNYAN, H.L. 'Experimental Investigations of the SEWALL,

J.L

Effects of

Concentrated

Weights on

the

Flutter Characteristics of a Straight C Cantilever Wing' NACA TN 1594, 1948. RUNYAN,E.L. 'Flutter of a Uniform Wing with an WATKINS,C.E. Arbitrarily Placed Concentrated Mass

(13)

and a Coaparison with Experiment' NACA Rep. No. 966

.8. RODDEN,W.P. _{'A Matrix Approach to Flutter Analysis'}

SMP Fund Paper No. PP-63, Inst. of the Acre. Sci., May 1959.

RODDkN,W.P. 'Flutter and Vibration Analysis by a PARXAS,E.P. Collacation Method: Analytical Develop-MALCOM,H.Â. ment and Computational Procedure' Rep.

No. TDR 169 (3230-11), TN-14, Aero-Space Corporation, Calif., July 1963. GLLNITZ,H. 'Calculation of Eigenvalues and Eigen-GOLLNITZ,H. vectors of Large NDn-Hermitian Matrices' WILLE,F. In AGARD Rep. No.511, NATO,AGARD.

(14)

0.0

000

0.0

0.0 .001670

-.000599

.008476

.004711

.002442

.024273

0002442

.011517

.018223

.042424

.007143

.004874

.041570

.035520

.004874

.013949

0035520

.059721

.009576

0007307

0058868

.052018

.007307

.016382

.052818

.077019

.012616

.010347

.080489

.074439

.010347

.019422

.074439

.098640

.079653

.070578

.106880

.118572

.109V

.109497

.183O

.183555

.158145

.194447

O

186924

_.174823

.223227

.273411

.261310

.420338

.261310

.309714

.404456

.467895

TABLE i b) STRUCTURAIJ INFLUENCE C0EFICIENT MtTRIX

(in./ib.)

.124376 -.005433 0.0

0.0

0.0 -.005433

.084728 0.0

0.0

0.0 .082917 ..00662 0,0

0,0

0.0 -.003662

.056486 0.0

0.0

000

0.0 .08291? v.003662 0,0

0.0

0.0 -.003662

.056486 0.0

000

0.0

0.0 .08291? -.003662 0.0

0.0

000

0.0

0.0 -.003662

.056486 0.0

0.0

000

0.0

0.0 .124376 -.005433

000

0.0

000

0.0

000 -.005433

.084728

TABLE i a) Nasß

for tho Model

j1b.)

0.0

0.0 0,0

0.0

000

0.0

(15)

All the other elemente are equal to zero.

TABLE 1. c) Ele.ets of th

_{Arodynaic Influenec}

Co cff ici ent Uatrix

a14

a1,2 a2, = a9,9 = a9,10 = a10,9 = (1.252620 - 0.273872) =(-1.27475 + i 0.133870) (0.000938 + i 0.075116) &2,2 = a10,10= (0.002816 - 1 0.075116) a3,3 = a595 ₌ _{= (0.835078 - i 0.182582)} a. = a- = a 7,8 = (-0.849830 + i 0.089246) a 4, = a - = e.8,7 = .000616 + i 0.050076)