Nonlinear flutter of composite laminated plates

0895-7177/90$3.00 + 0.00

Math/ Comput. Modelling, Vol. 14, pp. 983-988, 1990 Printed in Great Britain NONLINEAR

L.C. Shiau

FLUTTER

and

Pergamon Press plc

OF COMPOSITE

LAMINATED

PLATES

L.T. Lu

Institute of Aeronautics and Astronautics National Cheng Kung University Tainan,

Taiwan,

R.O.C.

The nonlinear flutter behavior of a two-dimensional simply supported composAbstract. ite laminated plate at high supersonic Mach number has been investigated. Von Karman’s large deflection Galerkin’s

plate theory

method

linear ordinary integration damping,

differential

method. in-plane

equations

Kevwords.

aerodynamic

theory

have been employed. to a system of non-

in time which are then solved by a direct numerical

Nonlinear flutter results are presented with the effects of aerodynamic force, static

show that the anisotropic have significant

and qusai-steady

has been used to reduce the governing equations

pressure

properties

differential,

and anisotropic

such as fiber orientation

preperties.

and elastic

effects on the behavior of both limit cycle oscillation

Nonlinear

cycle oscillation,

flutter,

chaotic

quasi-steady

aerodynamic

Results

modulus

and chaotic

ratio

motion.

theory, large deformation,

limit

motion.

results

INTRODUCTION

show that the aerodynamic

damping,

forces, and static pressure differentials Because

of their superior

stiffness-to-weight materials,

strength-to-weight

materials

industries

craft structures

for the purpose of weight saving.

to use them efficiently and dynamic At current

itary aircraft,

behaviors

composite

laminated

plates are of particular

speed flight, the external exhibit

flutter.

and aerodynamic pressure stable

indicates

and grows exponentially oscillations,

amplitude

as dynamic

pressure

derstanding

and cannot

oscillation

itself.

effects

and very limited literature

flutter problems,

assesment

the geometrically

to large amplitude

oscillations

isotropic panels.

un-

composite

materi-

on this subject has appeared

for the case of nonlinear

[13] in 1967 used Rayleigh-Ritz

in the vicinity ary conditions.

flut-

method

panels with various

With panel aspect ratio equals 3,

his results show that the highest flutter boundaries

fall

of filament angle of 30” for all boundIn 1974, Rossettos and Tong (141 an-

alyzed the flutter of cantilever anisotropic plates by finite element method and the results indicate a gener-

have

behavior

in

Hence, assum-

are still not well defined

to study the flutter of orthotropic

nonlinear effect due

flutter

Ketter

especially

boundary conditions.

should be considered.

the nonlinear

Galerkin’s

ter.

of the panel

In the past thirty years, different approaches been used to study

in journals

about

For a more thorough

and more realistic

loads the motion

cases.

For panels made of advanced als, the flutter characteristics

the flut-

give any information

chaotic

pressure.

results for certain

tensile

increases.

Hence, the linear theory can only determine

may become

range of aerodynamic

lead to correct

un-

However,

nonlinear

the

of limit cycle oscil-

ing the flutter motion of the panel is harmonic may not

dynamic

the in-plane

prob-

However,

In 1982, Dowell [12] pointed out in his paper

certain

Linear strucbecomes

were limited to harmonic

limit cycle oscillation

elastic,

the panel motion to a bonded value with

ter boundary the flutter

lation.

panel flutter

plates.

that in the presence of in-plane compressive

oscillation

with time.

induced by the geometrically

increasing

solutions

During high

that there is a critical

above which the panel motion

will restrain

lems of 2-D and 3-D isotropic

Hence

of inertia,

forces of the system.

with large amplitude stresses

instability

IS-111 in late 1970s and

early 1980s to study the nonlinear

skin panel of an airframe may

Panel flutter is a self-excited

and is due to dynamic tural theory

interest.

agree well with those ob-

ployed by several researchers

mil-

are mostly used to

make skin of wings and fuselage of an aircraft.

the character-

and results obtained

tained by direct integration method. Because of its versatile applicability, finite element technique was em-

under various loads

materials

Some re-

balance method

method to determine

for 2-D and 3-D problems

But

of their

stage, in high performance

of the plate.

istics of the limit cycle oscillation

to replace matels in the air-

a good understanding

behavior

14-71 later on used harmonic

and perturbation

have been widely used

in aeronautical

is needed.

searchers

ratios, as compared with conventional

composite

structural

effect on the flutter

and

in-plane

have significant

ally strong lack of monotonic dependence on filament angle. More detailed studies about the linear flutter be-

of

method with direct numer-

havior of composite

ical integration was employed by Dowel1 [l-3] to study 2-D and 3-D nonlinear flutter of isotropic panels. His

laminated plates were performed in

Refs 15 and 16 using either Galerkin’s 983

method or finite

Proc. 7th Int. Conf on Mathematical and Computer Modelling

984 element method. sequency, gularity

Effects

anisotropic

of fiber orientation,

property,

were considered

show that anisotropic on the flutter

aspect

in their analyses

properties

rect numerical the nonlinear

integration

(1) where

is used to study

composite

are employed for the analysis. behavior motion

modulus

ratio;

pressure

differential;

modulus

ratio

All = 2

on

vi:’

plate are exam-

of fiber orientation;

pressure;

inplane

load; static

and mass ratio.

In Eqs.(3) rigidity $k)

a, thickness

a 2-dimensional 1.

The laminate

anisotropic

stacking

E:

_

COS4ek

+

~IZEZEI

2( El

M, and aerodynamic

pressure

to consist

-

“f2E2

load Nia) differential

air-

U, Mach number

Ap is assumed

is also subjected

passing x-

to an applied

and a geometrically

duced in-plane force iv!“). In addition,

Sin’

ok

CO.?

ok

El&

+ El

-

Sin*

nonlinear

in-

there is a static

P, across the plate.

ok

42E2

is symmet-

Supersonic

over the top surface of the plate along the positive The plate

+ 2G12)

sheets bonded

sequence

of the plate.

flow with air density p, flow velocity

pressure

in

for the k th layer and can be expressed

‘I - E, -ufzEz

thin plate with length

The plate is assumed

rical about the midplane

in-plane

rigidity.

reduced stiffness coefficient

h, and mass density per unit volumn P,,,

as shown in Fig.

direction.

is the plate bending

as

FORMULATIONS

of K layers of homogeneous together.

zk--1)

is the plate extensional

11 IS the transformed

$W Consider

-

and (4), the Dll

and the All

the x-direction EQUATION

(zk

k=l

for limit cycle oscillation

as functions

dynamic

theory

The effects of compos-

of the composite

are presented

plate

large deflec-

aerodynamic

angle and orthotropic

Results

sim-

laminated

flow. Von Karman’s

tion plate theory and qusai-steady

and chaotic

effect

method coupled with di-

technique

symmetric

exposed to supersonic

ite filament

$ - (NL") +NL')) g +p,,,hg+Ap =P,

Dll

and results

have significant

flutter behavior of a two-dimensional

ply supported

ined.

flow an-

boundaries.

In this paper, Galerkin’s

the flutter

stacking

ratio,

(5) in which El, Ez, Glz, and VIZ are the engineering constants, and gk is the angle between fiber direction and the x-axis. For high Mach numbers namic pressure dimensional

(A4 > 1.6) the aerody-

loading Ap is assumed

quasi-steady

supersonic

to follow two-

aerodynamic

the-

ory: 1 aw

M2-2

.?.w

5,424~~

Introducing

the following nondimensional

1 (6) param-

eters and constants

W

-e=r

= w/h

(

=zfa

T

(a)

Plate

conflguratlon

P

=s 11 (7)

Fig, 1 Plate geometry The

differential

and nlv stacking

equation

of motion

sequence. for this 2-D

composite plate with large oscillations can be described by von Karman’s large deflection plate theory as

Eqs.

(1) to (6) can be nondimensionalized

bined into one equation

and com-

as

SW

--

w

R I

985

Proc. 7th Int. Conf on Mathematical and Computer Modelling

layers of lamina and the fiber angle of the top layer is defined as +B. where 0::’

is the value of Dll

aligned

with the z-axis

m

has been reduced to 6 Eq.

when all fibers

and the expression

are

in Eq. (8) for M >> 1.

(8) will be solved by the use of Galerkin’s

method.

For a simply supported

ment function

plate,

the displace-

is assumed to be

w(t,r)

Substituting

Convergence

study

(e)2 When using series solutions determine

it is of importance

obtain converged results. convergence

Fig. 2 shows the results of a

study for symmetric

angle-ply plates with

different fiber angles. The limit cycle oscillation

= 2 u,(r)sinnnE n=,

(9)

to

if sufficient terms are used in the analysis to

ampli-

tudes are plotted vs. number of modes used. For small angle of fiber orientations are sufficient

(0 = 0” - 500), 4 to 5 modes

to give accurate

results.

For larger fiber

angles (0 2 60”), 6 to 7 modes are needed. The modes

Eq. (9) into Eq. (E), yields

required for convergence is also dependent on the magnitude of the aerodynamic pressure on the plate aa shown in Fig.

2(b).

namic pressure to obtain

When 0 = 70”, at low aerody-

(X = 50 or loo),

converged solution.

6 modes is enough

However, at X = 200 or

higher, more modes are required. In this study, six modes are used in all calculations.

Satisfactory

results are expected

to be obtained

for fiber angles larger than 60” since the aerodynamic + R,Cu,(n7r)2sinn~<+ n

C $sinn7rc Il

pressure considered

is low for B > 60” in order to keep

the plate amplitude

within certain

range.

=P Multiplying tegrating

both sides of Eq.

(10) by sinsat

and in-

over the panel length, one obtains

NUKRER

YODR

(b)

+

f(L)”

!!!!!I = 11-s(;wp l,...,cc

5 =

Eq. (11) represents nonlinear, mination

(11)

ordinary

a coupled set of second-order, for the det,er-

different~ial eqwhons

of the time history of the modal amplitudes.

The anisotropic

properties

of the 2-D composite

plate

are included in the Eq. (11) through variables Dll and All.

RESULTS The numerical formed on VAX-8600 posite material

AND DISCUSSIONS integration

process

Effect of aerodynamic

The com-

used in this study is Graphite/Epoxy

AS-3002 with El/E2 = 26.5, Glz = 1.184 Ez, and ~12 = 0.21. The laminated plate is composed of 10

studv ff = 0.75,

damping

The effect of aerodynamic amplitude

damping

on the plate

of the limit cycle for different fiber orienta-

tions is given in Fig. 3. Aerodynamic damping has less effect on the oscillation amplitude as the fiber angle increases.

However, for small fiber angles and high

aerodynamic

pressures,

pronounced

has been per-

high speed computer.

Fig. 2 Amplitude convergence /L/M = 0.01).

the aerodynamic

damping has

effect on the plate amplitude.

Hence, for

design purposes, the range of aerodynamic damping coefficient (or mass ratio) should be determined precisely in order to get lower amplitudes

which in turn

give lower stresses in the plate. Nevertheless, the aerodynamic damping should be included in the numerical time integration

analysis

if the limit cycle oscillation

986

Proc. 7th Int. Conf. on Mathematical and Computer Mode&g

Fig. 3 Limit cycle amplitude

Fig. 5 Limit cvcle amulitude

vs dynamic pressure

(f = 0.75.ulM

(( = 0.75).

and without

of a 10” symmetric considering

angle-ply plate with

aerodynamic

damping.

clear seen that the plate response without aerodynamic

damping

pressure

Fig. 4 shows the time his-

is going to be reached. tory responses

vs dvnamic

= 0.01).

(by setting

reach the limit cycle oscillation

considering

10-

6

p/M = 0) will never

and is strongly

dent on the given initial disturbance. sponse including aerodynamic

It is

depen-

For the plate re-

damping, the limit cycle

is reached at 7 = 4.0 and is independent

of the form of

the initial disturbance. n d

g fi

1

0

I

10

0

I

I

60

50

40

1

70

I D

8

p/JY=o.o1

Fig. 6 Limit cvcle amnhtude (t = 0.75,~IM

vs dynamic oressure

= 0.01).

reason is that the plate amplitude dynamic

p//y=o.o

I

I

I

30

20

tensile

pressure stresses

increases

with the

which in turn gives higher in-plane

in the plate.

Higher in-plane

stresses mean higher restoring

tensile

forces in the plate which

make the plate vibrating

faster.

the limit cycle frequency

is about the same for all fiber

The increasing

rate of

orientations. Fig. 4 Time history

resoonse of nlate (13= lo“, Effect of elastic modulus ratio

x = 350. < = 0.75). Effect of fiber orientation

Fig.

7 shows the effect of elastic

on the amplitude The effects of fiber orientation and frequency in Figs.

on the amplitude

of the limit cycle oscillation

5 and 6. As expected

with the dynamic

pressure

are depicted

the amplitudes

increase

for all fiber orientations.

However, due to directional stiffness effect, i.e., rotating fibers away from the x-axis results in a reduction of the plate stiffness increasing

in the x-direction,

rate is proportional pressure

ratio fiber

angles. Increasing the modulus ratio increases the stiffness of the plate which decreases the amplitude of oscillation.

However, because

in the x-direction

the component

of El/E2

becomes smaller as the fiber rotating

away from the x-axis,

the reduction

rate of the ampli-

tude is smaller with higher fiber angle.

the amplitude

to the fiber angle.

The frequency of the limit cycle oscillation with dynamic

modulus

of the limit cycle for different

Effect of static

pressure differential

increases

for all fiber orientations.

The

The effect of static pressure differential

can be eas-

Proc. 7th Int. Co& on Mathemafical and Computer Modelling The interesting and

,

1.8

I

chaotic plane,

occur

can earily A in Fig.

dynamic

pressure

12

,

14

16

I

ily seen from phase the presnece

20

diagram

+W

pressure

decreases

direction.

symmetry

plane plot as shown

of static

22

with

respect

phase

R, is further shown

10(c).

for certain

parameter random.

of displacement

10

the motion

-50

becomes

chaotic

which

steady

of X and R, might

the minima

and maxima

of the motion

are bounded

150 /

equilibrium Umlt

r<

cycle oacihaion

Flat and atable

10

SO-

t=o.75

-to

-0.5

w

-50 -,.s

I.5

0.5

-0.1

Buckled

1.5

0.5

0 0.0

0.5

plate

1.0

(2) 1.5

2.b

-Rx’

w

30

50

10

10 -10 -50 -50

k

-1.5

-0.5

1.5

0.5

w

(a)

Fig. 9 Stabilitv

-10 -SO -10 -1.8 -0.5 0.5 1.3

P=ZOO

Fig. 8 Phase

plane

regions

I[ = 0.75. u/M

(b)%=O

10

to -10

-50

plot (9 = 20”. X = 500,

I-LA 0.5

-0.5

-1.0

p/M = 0.01).

1.5

w

(4 15

of in-plane

= 0.01).

(=0.55

-PO

(b)

,

I

loading 0

The effect of in-plane

loading

aries was studied.

The results

sure

compression plate motion

X vs in-plane The

for 0 = 40”. be divided

into

four

types:

the plate

remains

moderate

R,, the plate

ble situation, oscillation monic just

stable

motion,

(1) for small

and

flat,

buckles

(4) for sufficiently occurs.

bound-

as dyamic

pres-

are shown in Fig. 9 which may occur can

but with simple

the boundary

motion

on the flutter

plotted

(3) for moderate occurs

above

chaotic

as

occurs

50

50

the

As the

the

is no longer

Eil E@l I33!iiH

-1.5

Effect

in Fig. and

plane.

-30

-50

motion

motion

combinations However,

as shown

50

c

-10

.4s the

the whole towards

50

SO

10(a).

with buckling.

The chaotic

and velocity

in the phase

of

motion

9, the limit cycle oscil-

is the flutter

loo-

50

t

increased,

in Fig.

harmonic

motion

are associated

be termed

position.

k

B in Fig.

orbit

orbits

may

types

from 120 to 60 but R, remains

a subharmonic

larger

three

in Fig.

in the

which

in Fig. 8. With

diagram

to its mean

24

ratio

differential,

in size and shifts

Also the

I

I

1

Fin. 7 Limit cvcle amulitude vs modulus (X = 350, f = 0.75. h/M = 0.01).

phase

The

shows

simple

9 is depicted drops

i.e., point

becomes

are plotted

of motion

Fig.10

of point

two smaller I

be seen.

The limit cycle with

IO(b).

I

type

motion.

lation

0.0

If the results

the distinctive

unchanged,

0.3 -

csses are the limit cycle oscillation

motion.

phase

987

A and

(2) for small

X and

sta-

R,, limit cycle

harmonic

or subhar-

high R, and the X is

of the dynamic

stable

(c)

-0 -15

-,.5

-0.5

R,,

X and

but is in dynamic

l O

region,

Fia. 10 Phase

W

0.1

plane

1.5

plot of points

in Fig.

9.

Proc.

988

7th Int. Conf

on Mathematical

and Computer

12. Dowel], E.H.,

CONCLUSIONS

Example The

nonlinear

sional composite

flutter

behavior

laminated

of a two dimen-

plate is demonstrated.

parameters

studied include fiber orientation,

ulus ratio,

areodynamic

ferential,

damping,

static

and in-plane compressive

load.

The aerodynamic on the amplitude

The

elastic modpressure

dif-

oscillation

effect

for small

fiber angles and has less effect for larger fiber angles. behaviors

has significant

such as flutter

location

of the maximum

limit cycle or chaotic laminated

effect on all flutter

frequency,

flutter

amplitude

vibration.

amplitude,

of the plate,

Properly

plate can improve the fluitter

reduce the flutter

of Chaotic

Autonomous

tion, 1982, 85(3), 13. Ketter,

D.J.,

Panels,”

and

design the stability

and

amplitude.

14. Rossettos,

REFERENCES “Nonlinear

Oscillations

of a Flutter-

Vol. 4, July

1966, pp.

1267-1275. 2. Dowell, E.H., ing Plate

“Nonlinear

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Vol.

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1967,

pp. 1856-1862. 3. Ventres, Theory

C.S. and Dowell, E.H., and Experiment

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“ Comparison

for Nonlinear

AIAA Journal,

Flutter

of of

Vol. 8, Nov. 1970,

pp. 2022-2030. 4. Eastep,

F.E. and McIntosh,

linear Panel Flutter Excitation

or Nonlinear

AIAA Journal, 5. Morino,

S.C., “Analysis of Non-

and Response

L., “A Perturbation

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Panel

under Random

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Problems,”

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of

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AIAA Journal,

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1977, pp. 1107-1110. 9. Rao,

K.S.

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G.V.,

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1967, pp.

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l’ancl

AlAI\ .lour-

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1075-1080. 15. Sawyer,

J.W.,

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Plates,”

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L.C.,

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116~124.

damping has pronounced

of limit cycle

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Modelling

D.H.,

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