Large deflection panel flutter

LARGE DEFLECTION PANEL FLUTTER

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE REcrOR MAGNIFICUS Dr. R. KRONIG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 24 JANUARI 1962, DES NAMIDDAGS TE 2 UUR

door

EDMOND FRITSEMILE ZEIJOEL

Vliegtuigbouwkundig Ingenieur geboren te Modjokerto, Java

Dit proefschrift is goedgekeurd door de promotor Prof. Dr. Ir. A. van der Neut

AAN NIJN MOEDER

De schrijver van dit proefschrift betuigt hiermede zijn dank aan

het bestuur van het "Air Force Office of Scientific Research", Air Research and Deve::.opnent Command, in Washington, D. C., voor de toestemming, resul-taten van onderzoekingen, welke onder het contract No. AF 49(638)-389 zijn uitgevoerd, in dit proefschrift op te nemen.

de directie van het "Midwest Research Institute", in Kansas City,. voor de vrijheid om de aan hem opgedragen onderzoekingen de voor dit proefschrift wenselijk geachte vorm en afwerking te geven.

Mr. D.I. Sommerville voor het uitvoeren van de in dit proefschrift voor-komende numerieke berekeningen op een automatische rekenmachine.

'l'ABLE OF CONTEWrS

List of Symbols • •

.

.

.

.

1. Introduction

. . .

.

. .

. .

. . . .

.

. .

.

.

.

. . .

H. Equations of Motion of Large Amplitude Vibrations of Plates •

.

. . .

.

. .

. . .

.

A. Nonlinear Plate Theory of Elasticity

B. Work of External Forces • C. Work of Inertia Forces D. Equations of Motion • •

E. Formulation of Boundary Conditions

111. Aerodynamic Forces

.

. . .

A. Unsteady Supersonic Aerodynamic Theory

B. Equations for Nearly Flat Plates

.

C. Solution of Potential and Pressure Equations for

an Array of Rectangular Panels with Simply Supported Edges • • • • . • . • • •

. .

IV. Simplification of Large Amplitude Flutter Equations for an

V.

A. B.

C.

Array of Rectangular Plates • •

Initial Simplifications of Equations

.

.

.

.

.

.

.

Simplified Representation of Membrane Forces Due to

Large Amplitude Deflection for Rectangular Panels with Simply Supported Edges • • • • • • . • . • • Flutter Equations Utilizing Simplified Membrane Forces

for an Array of Rectangular Panels with Simply Supported Edges • • • • • • • • • • Method for Determination of Flutter Conditions

.

Page No. viii 1 4 4 15 15 16 20 23 23 26 31 41 41 44 69 75

A. Parameters in Flutter Equations • • • • • • . • . • •• 75

B. First Method for Determination of Flutter Conditions 76

VI.

TABLE OF COJIlTENTS (Continued)

Numerical Results and Discussion • • A.

B. C.

Small Deflection Theory, Q~v ;::; 0 Large Deflection Theory, Q~v

f

0 • Concluding Remarks •

References • • • . ••

Appendix (Figs. 1-20, Tables I-XVIII) Beknopt Overzicht Figure 1 2 3 4 5 6 7 8 LIST OF ILLUSTRATIONS Array of Panels

Stability Boundaries from Two-l·iode Analysis for Certain 1'-1, s , g , P Y ;

1

== 1 • • • • . • • • . {Af ter Fung [27J }: Small Deflection Stabili ty

Boundaries for Aluminum Panels at Sea Level, s == P == 0 ,

I

=

1 . Two-Dimensional Flow, except for Finlte Aspect Ratio Panels, which are Eased on Aerodynamic

"Strip 11 Theory from

[5]. . . .

. . . .

~-k Graph for Pinned-Edge Aluminum Panels at Sea Level from ~our-Mode Analysis, 1-1 = 1.3 , g

= s

==

cr:

= 0;

J!.

= 1 (Table I) • • • • • • • • T-k Craph for Pinned-Edge Aluminum Panels at Sea

Level from ~ix-Mode Analysis, M ;::; 1.3 , g ;::; s

=

(), = 0;

P

= 1 (Table 11) . • • • • • • • • •

Small Deflection Stability Boundary from Four- and

One-Mode Analysis for Pinned-Edge Aluminum Panels at Sea Level, g

= s

= o-y = 0 ;

Z;::;

1 (Table lIl) . Small Deflection Stability Boundaries from Four-Mode

Analysis for Pinned-Edge Alwninum Panels at Sea Level, g

=

0.01; o-y

=

s ;::; 0 ;

L=

1 (Table V) . . Small Deflection Stability Boundaries from Four-Mode

Analysis for Pinned-Edge Aluminum Panels at Sea

Page No. 80 81 86 88 90 93 125 Page No. 94 95 96 97 98 99 100

Figure 9 10 11 12 13 14 15 16 17 18 19

TABLE OF CONTENTS (Continued) LIST OF ILLUSTRATIONS (Continued)

Critical Panel Thickness Ratio versus Structural Damping for Pinned-Edge Aluminum Panels at Sea Level s = (). = 0 . ,- = 1 _{, y ' ' a v} Q , = 0

(Table VII) . . . .

Small Deflection Stability Boundaries from Four-Mode Analysis for Pinned-Edge Aluminum Panels at Sea Level, g = 0': = 0 , s = 1/4 ;

Z = 1 (Table VIII)

Small DeflectioK Stability Boundaries from Four-Mode

Analysis for Pinned-Edge Aluminum Panels at Sea Level, g:: 0: = 0 ; s = 1/2,

.I

= 1 (Table IX) Critical Panel

~hickness

Ratio versus Inverse of

Aspect Ratio for Pinned-Edge Aluminum Panels at Sea Level, g = o-y = 0;

f::::

1 (Table X) • • • • Small Deflection Stability Boundaries from Four-Mode

Analysis for Pinned-Edge A1UIninum Panels at Sea

Level, g ::

cr

y = s = 0 j ; [ = 2 (Table XI) . • . . • Small Deflection Stability Boundaries from Four-Mode

Analysis for Pinned-Edge Aluminum Panels at Sea Level, g ::::

oy

= S = 0;

.f.

= 3

(Table XII) • • •

Critical Panel Thicy~ess Ratio versus Number of Panels in the Chordwise Direction from First-Mode Boundary for Pinned-Edge Aluminum Panels at Sea

Level, (Jy:: 0 , Q~v = 0 (Table XIII) • . • • • • • . Small and Large Deflection Stability Boundaries from

Cne-Mode Analyses for Pinned-Edge Aluminum Panels at Sea Level, g = O. Ol ; s

=

1/4 ;

f

= 1 Page No. 102 103 104 105 106 107 108 (Table XIV) • • • . • • • • • • • • • • • • • • • 109 Large Deflection Stability Boundaries from One-Mode

Analysis for Pinned-Edge Aluminum Panel~ at Sea Level, g

=

O. Ol , s = 1/4 , o-y = 0,

-l

= 2

(TabIe XV) . . . .

Large Deflection Stability Boundaries from One-Mode Analysis for Pinned-Edge Aluminum Panel~ at Sea Level, g :::: O. Ol , s = 1/4, (Jy

=

0 ,

L

= 3

(Tabie XVI) • • • • • • • • • • • • • • • • Panel Thickness Ratio versus Maximum Stress from

Cne-Mode Analysis for Pinned-Edge Aluminum Panels

110

Figure 20 Table I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII

TABIE OF CONTENTS (Continued) LIST OF ILLUSTRATIONS (Concluded)

Amplitude of Vibration/Plate Thickness versus Maximum Stress from One-Mode Analysis for Pinned-Edge Aluminum Panels at Sea Level, g

=

0.01 ;

s

=

1/4 ;

cr..

= 0 ; Q~v

f

0 ; E = 1. OS x 107 lb/ in2 (Table

XVIII~

. • • • . • • • • • • • • • • • • • • • •

LIST OF TABLES

Flutter Vector, Thickness Ratio and Reduced Fre-quency Corresponding to Four-Mode Analysis at M

= 1.

3 (see Fig. 4) . . • • • • • • • • • • • Flutter Vector, Thickness Ratio and Reduced

Fre-quency Corresponding to Six-Mode Analysis at

M

=

1.3 (see Fig. S) • • • • • • • • • • • • • • • • • Panel Thickness Ratio, 'T (see Fig. 6). • • • • • • • • Critical Flutter Vector, Thickness Ratio, and Reduced

Frequency at Various Mach Numbers Corresponding to Four-Mode Analysis (see Fig. 6) •• • • • • • • • Panel Thickness Ratio, t (see Fig. 7). • • • • • • • • Panel Thickness Ratio, T (see Fig. 8) . • • .

Critical Panel Thickness Ratio, icr (see Fig. 9) Panel Thickness Ratio, T (see Fig. 10) . • • Panel Thickness Ratio, T (see Fig. 11) • . •

Critical Panel Thickness Ratio, T (see Fig. 12) • cr

Panel Thickness Ratio, y (see Fig. 13) • • • Panel Thickness Ratio, T (see Fig. 14) • • • Critical Panel Thickness Ratio, ~cr (see Fig. lS) • Panel Thickness Ratio, t (see Fig. 16) •

Panel Thickness Ratio, T (see Fig. 17) . • • Panel Thiclmess Ratio, T (see Fig. 18) • . • • Panel Thiclmess Ratio, t', versus Maximum Stress, (~ t t 1) _{xx oa max} ,lb/in2 (see Fig. 19) • • • • Amplitude of Vibration/Plate Thickness versus

Maximum Stress, (~xx total)max' lb/in2 • • • . • • . •

Page No. 113 Page No. 114 11S 116 117 118 118 119 119 120 120 121 121 122 122 123 123 124 124

a

b

LIST OF SYMBOIS

= effective cross-sectional area of supporting framework in

x-direction

= effective cross-sectional area of supporting framework in

y-direction

=

panel chord = defined by (5.56)

= defined by (4.16)

= panel span

= defined by (4.16)

=

free stream velocity of wave propagation in air

= velocity of wave propagation of longitudinal vibrations

in plates (see (4.68»

Cr m,dr m,er m,fr m

,

,

_,

,

=

defined by (5.60)-(5.65) D

_{= - - - -}

Eh5

12(1- 1)2)

= flexural rigidity of plate

= defined by (4.16)

E

₌

modulus of elasticity

E

_{= modulus of elasticity in shear}

2(1+ V)

= defined by (5.64) and (5.65), respectively

F

=

stress function defined by (4.6)

= chordwise downwash distribution

f

_{= arbitrary function representing a source strength}

= chordwise deflection function

G(y) g g(y)

~"m"m

h

hr,m"m

K k

.p

,m M

Mx,My,,!>1

xy n p(x"y"t)

LIST OF SYMBOLS (Continued)

=

spanwise downwash distribution

= defined by (3.73)

~ coefficient of structural damping

= spanwise def1ection function

= defined by (3.15)

= defined by (3.14)

=

panel thickness

= defined by (3.16)

= defined by (3.66)

= Bessel fllnction of first kirJ.d of order n

=

-1 _Mk

--(32

=

~

= reduced frequency

U

=

w1a

=

1(2

1-1....

U

-Ji2

M /t2

= number of panels in chordwise direction (see Fig. 1)

=

direction cosines defined by (2.16)

= Mach nuffiber

=

~

coo

=

moment per unit length defined by (2.20)

=

cb.ord~t1ise modal n1.illJ.ber

=

force per unit length defined by (2.20)

=

spanwise moda1 number

Q q ~,n R ~,n r S s T t

u

ü,v,w

u,v,w Vi W(x,y, t) W

LIST OF SYMBOLS (Continued)

=

perturbation pressure at upper surface of plate

=

defined by

(3.44)

=

defined by

(4.47)

=

average value of Q in time

=

integer

=

generalized coordinate of m nth

,

mode shape

=

defined by

(3.10)

=

defined by

(4.37)

=

defined by

(4.49 )

=

integer

=

integration area defined by (3.11)

=

!

=

~

=

inverse of aspect ratio of panel

b

AR

:; 'kinetic energy

=

initial membrane forces per unit length

=

time

=

free stream velocity

~ displacement components in x-, y- and z-direction

=

displacement camponents of middle surface of plate in x-, y- and z-direction

=

i th flutter vector

=

downwash

x,y,z x',y',z'

r,r

f-xx'€YY'€zz } f- xz ' €xy' f-yz

~m,m

p

Ps

LIST OF SYMBOLS (Continued)

= eoordinates of plate partiele

=

nondimensipnal eoordinates

=

defined by (4.23)

=

~M2_1

=

defined by (3.50) and (3.39), respeetive1y

=

defined on p. 72

=

eomponents of strain tensor

=

defined by (3.72)

=

defined by (3.71)

=

defined by (3.55)

=

frequeney of vibration

=

wM2

Uf32

=

mass air density

=

mass density of panel

=

buekling stress in spanwise direetion

=

ehordwise and spanwise eoordinates, respeetively

= veloeity potentia1

=

~

=

nondimensiona1 panel thiekness a

=

value of 't" for whieh

%

= 0 , DI = 0

=

delay times defined by (3.15)

=

eomponents of symmetrie stress tensor

LIST OF SYMBOLS (Concluded)

=

~

= modified maas density ratio

Is'

coo

= --- =

modified velocity of wave propagation ratio

~T

~(x,y), If'(x,y), x.(x,y) = defined by (2.3)

.

_J(

₎ ( )

=

; t ( )

,

=

dimensionless quantity I _)n

=

_{normalized coordinate} \

I ( ) I

=

modulus of complex ( )

I • . INTRODUCTION

In modern aircraft, a metal covering or "skin" is of ten utilized to obtain an aerodynamically "smooth" surface and thus prevent excessive drag. This skin is gene rally divided by stiffeners, ribs and other struc-tural members into rectangular fields, which may be considered as an array of rectangular plates supported along the edges. These plates are subjected to loads in their plane, since they are in the load path of the structure as a whole, and in addition to aerodynamic farces normal to their plane on the outer surface of the plate. It is weIl known that the mutual interacting be-tween elastic, aerodynamic and inertia forces on structural members of air-craft could lead to unstable conditions. The phenomenon of aeroelastic instabil1ty of aircraft skin panels leading to oscillatory motion of the panels is called "panel flutter".

Panel flutter has been the subject of a number of theoretical in-vestigations. In most of the early papers the authors, such as Hayes

[IJ,

Miles

[2J,

and Shen [5], treated the problem of a one-dimensional flat or

slightly curved panel of a finite chord length, held at its leading and trailing edges, with a supersonic flow on its upper surface. Hayes con-siders the dynamic stability of G buckled panel using steady-state

aerody-namic forces. Miles extended the investigations of Hayes by using quasi-stationary aerodynamic forces, which were derived by retaining terms up to the first power of the reduced frequency. He finds that all panels, re-gardless of thickness, are unstable for Mach numbers between 1 and

....J2 •

Using exact linearized aerodynamic forces Shen points out the necessity of including higher order frequency terms at the values of Mach number near

~ and stresses the importanee of three-dimensional effe cts. Further in-vestigations on one-dimensional panels were carried out by Nelson and

CUnningham

[4]

who indicate that at supersonic speeds beween Mach 1 and

42

a relatively large plate thickness (in the order of 1/100 of the panel chord) is required to prevent flutter instabili ty for aluminum panels. It is therefore of considerable importanee to examine more carefully the in-stab1lity characteristics of thin panels at supersenic speeds between Mach 1 and

...r2 •

More recently, the aeroelastic behavior of wo-dimensional panels in a three-dimensional flow has been treated by Eisley, Hedgepeth, Luke, and St. John. Eisley

[SJ

approximates the aerodynamic forces by using quasi-steady aerodynamic theory, while Hedgepeth

[6J

applies the so-called "statie" airforces by neglecting aerodynamic damping. Luke and st. John

[7J,

however, employ linearized three-dimensional aerodynamic theory and specialize to the case of an infinite span panel separated into rectangular bays by equally spa eed rigid stiffeners in the streamwise direction. A

comparison of the results of these investigations indicate the importanee of using three-dimensional aerodynamic theory in the region between Mach 1 and12 •

The analysis of a one-dimensional panel on many equally spaeed

sup-ports in two-dimensional supersonic flow has been performed by Hedgepeth

[sJ.

Rodden

[9J

considers a periodically supported infinite span panel and obtains an explicit stability boundary for a two-mode analysis. Miles and Rodden [lOJ define wo different types of instability, a single degree-of-freedom instability and a coupled mode instability, and point out that since Hedgepeth neglects aerodynamic damping his results concern only coupled mode instability. Al.though the results in [S] and [9] for coupled mode instability agree, it was found that at Mach 1.3 and t{2 , single

degree-of-freedom instability t~es place for which no stability boundary was

found. Miles again stresses the importance of using three-dimensional

aero-dynamic theory for the region beween Mach 1 and -\[2 •

In the analyses above, small deflection theary is applied to derive the aerodynamic and elastic properties of the configuration. Tbe resulting equations of motion are a set of linear differential equations, and the flutter speed is found from the boundary between stable and unstable

regions. In the stable region, the motion is damped while in the unstable

region the motien diverges. If the load carrying capacity of the configura-tion can adequately be described by linear theory, tbe unstable region

corresponds to failure of structural members. However, it is well known

tbat for thin plates large deflection theory must be used to determine the

ultimate load condition. Consequently, linear plate theory should not

with-out further investigation be used in panel flutter analysis if a criterion of failure is sought.

Large deflection plate theory 'fas applled to the panel flutter

problem for the first time by Fung [11]. Fung treats the problem of a

buckled plate in two-dimensional supersonic flow. Quasi-steady aerodynamic

farces were assumed for simplicity and the nonlinear equations solved by

tbe method of Kryloff and Bogoliuboff

[12].

Only two modes of vibration

were considered. More recently the same problem bas been discussed by

Shen [13J who solved the partial differential equation of motion by the same method without introducing generalized coordinates. These analyses are not

campletely satisfactory because no estimation of maximum stress is given.

Als 0, the problem has been restricted to a one-dimensional panel in

two-dimensional flow.

It is next of interest to examine the problem from a designer's point of view and to review the important conditions for the design of stable

which has to be investigated along with the necessary physica~ assumptions which need to be introduced.

Panel f~utter usuallY becomes the primary design criterion for those areas on the surface of supersonic airplanes and missiles which are not designed to carry appreciable structural leads. These areas are found on the tail surfaees of supersonic airplanes snd at many locations a~ong

missile bodies. Consequently, the array of rectangular panels can be initially flat or curved with smallor large aspect ratio. Also, initial membrane forces may be present due to statie aerodynamic loading and aero-dynamic heating.

On the tai~ surfaces of airplanes the arrays are approximately flat with small or ~arge aspect ratio. The aerodynamic lift on these sur-faces creates membrane forces in the spanwise direction which are either positive or negative. The statie pressure difference between the outside surface of the plate which is exposed to a supersonic stream and the inside surface of the plate leads to positive membrane forces in the spanwise and chordwise directions. Aerodynamic heating, however, often results in nega-tive membrane forces in both directions. When these latter forces are of sufficient magnitude, buckling occurs.

The array of panels on missile bodies are s~ightly curved with

sma~ aspect ratio. When pressurization is used, large positive membrane forces in the eireumferential direction can be expected. Statie deflection of these bodies, however, causes positive or negative membrane forces in the direction of flow.

Although the effects of initial curvature on panel flutter are of

eonsiderab~e importance [14], in this report the flutter analysis is restric-ted to approximately flat panels. The configuration studied consists of an infinite span plate whieh is separated into sn array of rectangular panels

(see Fig. ~). Since the region between Maeh 1 andi2 is of particular interest, ~inearized three-dimensional aerodynamic theory is applied. To obtain a criterion of failure, nonlinear plate theory is employed to describe the elastic behavior of the configuration. For simplieity, the usually beneficial effects of statie pressure difference between the upper and lower surfaees of the plate are neglected. Also, the dynamic pressure fluetuations of sti~ air on the inside surface of the plate is neglected.

Specia~zing to those cases with relatively large aspect ratios and perfect flexible supports in the chordwise direction, an estimation of maximum stress due to large deflection is giveA, Statie membrane forces in the spanwise direetion are also considered.

Numerical results are derived and the effects of number of degrees of freedom, aspect ratio, structural damping, number of panels in the chord-wise direction, statie membrane forces in the spanchord-wise direction and large deflection are examined.

Il. EQUATIONS OF MarION OF LARGE AMPLITUDE VIBRATIONS OF PLATES

A. Nonlinear Plate Theory of Elasticity

The basic equations of nonlinear plate theory are taken from the work of Novozhilov [15]. This theory has been applied by Herrmann [26J to derive the large amplitude equations of motion for plates. Of particular interest is the theory of elasticity of thin plates which are exposed to transverse loading. As in ordinary plate theory it is assumed that the fibers of the plate, which are perpendicular to the middle surface, remain perpendicular af ter deformation and that the shortest distance between each

point of the plate and the middle surface remains unchanged. In addition,

it is assumed that the elongations and shears, and also the angles of

rota-tion of the elements of the plate, are small compared to unity. Of special

interest is the case where the rotations in planes perpendicular to the plate are large with respect to the strains, which implies that the plate thickness is small since the strains and rotations are not independent. The

nonlinearity introduced by these assumptions is a geometrie one. Physieally

the plate material is considered linear whieh justifies the application of

Hooke's Law as a stress-strain relation.

1. Strain components and deflection: Consider a rectangular plate of constant thickness h. The Cartesian coordinate system is eon-sidered to be fixed, the x-y plane being the middle plane of the plate and the z-axis being directed normal to that plane.

Let the coordinates of a partiele of the plate before deformation

be x, y, and zand af ter deformation become x+u, y+v and z+w. From

the general three-dimensional theory of elasticity, the strain c·ompönents

----

---

-€xx

=

_dx Jü + ~ ₂

[(~~)2

+

(~:)2

+

(~:)2J

E

Jv

1

[(~;)2

+

(~;)2 +(~~)2J

=

+ -yy _{~y 2}

E

_ZZ

₌

- - + ~w

~[(~~)2

₊

(~:)2 +(;~)2J

Jz (2.1) Exy Jü ,)V

Ju

~ü

av

~v

aw

;w

=

- - + - - + - - + - - - - +

----~y dX dX ay ox ~y dX Jy

€

₌

.,> ü +

aw

+ Jü aü +

<Iv

dV

a

w aw

- - - - +

-XZ 6Z ()X dX OZ ox Jz

a

x ,) z

E

yz JV

aw

~ü

aü

av

Jij

J

w

JW

= --

_iJz + --- + - - - - + - - - - + --- - -_{Jy oy Jz Jy Jz Jy Jz}

Assuming that the fibers of the plate which were perpendicular to the middle surface remain perpendicular af ter deformation, and that the shortest di stance between each point of the p1ate and the middle surface remains unchanged,

E

₌

0 xz € _yz

₌

0 (2.2) €

₌

0 zz

Equations (2.1) and (2.2) suggest that the def1ections u, v and w be taken in the form

ü

=

u(x,y) + z ~(x,y)

ii

=

v(x,y) + Z \ti (x,y) (2.3)

w

=

w{x,y) + Z )(x,y)

where u, v and w denote the displacements of the middle surface of the plate.

Substituting (2.3) into the last three equations of (2.1) and utilizing (2.2), there is obtained

Ju

~+

(1

+

t..::!..)

\jJ +

a

w (l+X)

=

0

dy ay oy

(2.4)

_02 2 2

v- +

'+'

+ (1+ X) - 1

=

0

Solving for

7J-,

Y and

X.

and utilizing the assumption that the shears and elongations are small compared to unity, one obtains

_0

dW(

~v) 77":::::::-- 1 + -OX ay .Jv d W + -dX ay (2.5)

Substitutiog these values of

-#, '+'

and X ioto (2.3), the displacements

ü,

v

and ware obtained in terms of the displacements of the middle plane u, v and w. Because no restrietion is imposed on the

magnitude of the rotations of the e1ements of the p1ate, the displacements given by (2.3) and (2.5) correspond to streng bending of the p1ate. Frem the first three equations of (2.1) and (2.3) and (2.5), the general ex-pressions for the strain components ~ _xx , ~ _yy and E _xy may be derived

in terms of u, v and w. As may readi1y be verified, terms proportiona1 to z2 will appear in these strain components indicating a nenlinear vari-ation with p1ate thickness. For small e1ongvari-ations and shears, these term~

are smal1 and may therefore be neg1ected. Thus \

€

_xx

₌

_. ~ _xx + zV xx €yy

=

€ yy + ZVyy (2.6)

€

₌

€

+ z-V xy xy xy where (2.7) d \,/ ~ u iJ-7J- J v ,) '.J dW J X ).I =_'+'_+ _ _ + _ _ T_+ _ _

yy ~y iJy Jy Jy iJy ày ~y

=

J?J J\jI Ju Jij-

oU

r)?J-

dV Jy

+ + + +

-

-

-The parameters

é.

_xx

, €

_yy and

€

_xy are the elongations and

sbears of the middle surface, while tbe parameters "xx' vyy and v xy

cbaracterize the curvature of the middle surface.

When the angles of rotation of each plate element are small

com-PIU ';V

pared to un1ty, the derivatives _dX and --- may be considered to be of _r}y

tbe same order of magnitude as the elongations €xx

J u and

.:!..:!..

become of the same order of magoi tude

Jy dX

and

€

yy , while

as the shear

€

_xy

Moreover, for small plate deformations, the rotation of its elements about

the z-axis are small and second order terms containing such rotations may

therefore be neglected. The foregoing implies that higher order terms of tbe derivatives of u and v may be omitted in the _{- _} expressions for ~ , xx

€'yy' EXY' v_{xx ' vyy} and v xy Thus (2.7) becomes

E

₌

_---+ au

~(~)

2 xx

_ax

2 ~x

E

₌

.L!

+

~(~)2

yy ay 2 ay

€

= ~+ Jv

+

----

aw Jw (2.8) xy _ay Jx Jx Jy 1>

_{= --- =}

a7Y-

-

---

a

2w xx Jx _Jx2

v

_yy

₌

----

J'+' _Jx

-

J2w J _y2 J,J- _ay

₌

- 2 d 2W

v

₌

- - + xy Jy iJx Jx Jy

The approximate expressions for the strain components

€

_{xx '}

E:.

and

E:

thus become (see (2.6»

a u + 2

a

2w

E.

₌

!.(~)

- z ____ xx ax 2 ólx _d-~ Jv 2

a

2w

€

₌

_{- - +}

!.(~

₎

_{- z}

_--

(2.9) yy _{ax 2 tYy}

_a

y2

€

au l..:!..+ ;w dW _{- 2z}

a2w

=

- - +

-xy ay Jx dX ay q}i,. dy

The aJ)prox:imate displaecments

ü,

v

and

w,

correspond:ing to the strain components (2.9) may b-:: derived f!'om (2.3) and (2.5) by neg1ecting simi1ar terms of higher orde!'. Renee

u

₌

u

-

z ;}W Jx Jw _(2.10) v

₌

v

-

z

--dy

-w

=

w

These approximations a~~'e simi1ar to those proposed by van

Kármán

~~ for large def1eetions of plates.

2. Strain e~gy: The total strain energy of an e1astic body is given by

+ "t'

E

+ 1:'

E.

+ L

E. )

dxdydz

xy xy xz xz yz yz (2.11)

where the ~ij represent the symmetrie stress camponents taken per unit area.

The variation of the total strain energy is given by

+ 1:' erE, + T

JE

+ "c'

JE.

)dxdydz

xy xy xz xz yz yz (2.12)

Substituting the strain components given by (2.9) into the ex-pression for the variation of the tota1 strain energy (2.12) and uti1izing the approximation €xz

=

0 j €yz = 0 and €zz

=

0 , the variation of strain

energy for thin p1ates becames

(2.13)

Because the strains are written in terms of the strains of the middle surface of the plate, the integration with respect to the z-coordinate ean be performed.

h _+!:: N _x =

j~2

'l:xxdz ~=

!~2

"rxxzdz 2 2 +!:: +~ N =

1~2

_~yy dz _~=

1~2

1:: yyZdZ (2.14) y 2 2

j~~

+~

N = 7:_xy dz M

=

Jh

2

"C xy zdz xy xy

-

-2 2

Eq. (2.13) may be written as

+ N

{Ó

(~)+

b

(JV)+

~

Ó

(.7W) +

~

b

(~)}

+

xy Jy Jx Jx ay oy ox

(2.15)

Rea1i zing that 5 ( .:>'

U)

=

L

~

u,

S

(a

W)

=

~

bW , etc.,

OX Jx ax ax

the factors cf u, Ó v and ó w may be brought out by -partial integration obtaining

+

{L

(N

L.!

+ N

.1.:!)+..!.-

(N

~

+ N

~)

+

Ó>x , x Jx xy cJy cJy '\. xy rJx Y rJy

J~XY

- - " - + dXJy

_ {M

_{•. 'X} Q +M _xyTIl } cl _d

á

_x w _

{M

_xy

1.

+tLm}

_{._-y-} tY _dy

~

WJ

ds . +

(2.16)

in which

i

and mare the direct ion cosines of the normal to the boundary and the line integral is taken around all (external and internal) boundaries.

3. stress-strain and stress-displacement relations: Let the plate material be isotropic. In accordance with the assumption that the strains are small, the stress-strain relations become linear and obey Hooke's Law. Thus

EJO( _ 1

_{~

--y("C +'"t" )}

-

-E JO( yy zz

€

1 _{T yy --v(1: +T }}

=

-yy _E xx zz €zz

=

~

{LZZ

-)J ( --C::_xx+ 't:_{yy ) }}

(2.l7)

E.

=

2(1+ v~ txy xy _E

€

=

2~1+ }> ~ "1"xz xz _E

E.

=

2(1+ v~ ?: yz yz E

where E = the modulus of e1asticity in tension

E

2(1+V}

=

the modulus of elasticity in shear

1>

=

Poisson 's ratio

Solving for I':xx'""C and 1:' and neg1ecting the transverse

yy xy

norma1 stress, T_{zz and the transv.erse shear €xz , €yz} J just as in the

c1assical plate theory, gives

"l:"xx

=

E (E. +VE:, ) 1- 1.12 xx yy '"ryy

=

E (€yy+))E.xx ) (2.18) 1_V 2 "t' - E

€

xy - 2(1+1.» xy

Writing the stresses in terms of the displacements of the middle surface of the p1ate by using Eq. (2.9), one gets

1:"

= _

E

[{au

-

+ -1(JW)2 - + y -dV +")J-1(&W)2} - -z -

{d

+ ) ) -2 W J 2 w}] xx 1-V 2 a x 2 ... a x Jy 2 dy tJ x2 r)y2 "t"

= __ _

E [{rJV + -1 (èJW)2 - + y - +y-Ju 1(JW)2} - -z -

{c1

+ ' Y - • 2 W J2w}] (219) yy 1- 'V 2 r} Y 2 ;; Y r) x 2 J x

a

y2

a

x2

Combining (2.19) with (2.14) and performing the indicated inte~ gration over the p1ate thickness, there is obtained

N

=

6(1- V}D

[J

u + d v + ~ ~ ] xy h 2 a y ; x d x

a

y J 2w M xy = -D(l-V) - - -dX Jy (2.20)

where D is the plate modulus expressed as

(2.21)

These stress-displacement relations correspond with those obtained by ven Kármán [16

J.

B. Work of External Forces

The only external forces considered are aerodynamic and may there-fore be represented by forces acting in transverse direction of the plate. Let Pu denote aerodynamic pressure of the air flow at the side z> 0 of the platej then the virtual work done by Pu is given by

(2.22)

C. Work of Inertia Forces

For a body performing small oscillations about a position of equi-librium, the inertia forces per unit volume of a material of density l's are given by

..

- fs

w

where dote indicate differentiation with respect te time. The virtual work done by these force~ is given by

Now rep1ace Ü, v, and

w

by their approximate va1ues in terms of the def1ections of the midd1e surface as given by (2.10). The integra-tion with respect to z is easi1y performed and

+~

-fsl!!

h 2

{(ü-z

2 +

w

~

W } dxdydz (2.24)

h31p

a2.. ~2 .• +

f -

(---!

+ _ _

W)

Ó w dxdy + B 12 ~x2 ~y2 D. Equations of Motion

The equations of mot ion may be derived from the variationa1 equa-tion of small mot ion which states that during an arbitrary time interval the variation of the strain

ener~y

is equa1 to the work done by the externa1 forces and the inertia forces

Q..7J.

Thus,

Jw - áW_{e -} 5 T

=

0 (2.25)

a2.·

a2.·

a

2..

h3 "\2·· <J 2·· )

+ 1Y1x Nxy l'!.y ..

((7

w W +

ax2

+2 ax ay + dy2 -

fJrP

w+ Ps 12 ox2 + ay2

-

~u}]

ó"

dxdy +

(2.26)

+

f [{

Nxi +Nxy'" } $ u + {

Ny

i

+lIy'" } óv +

+

+

Equation (2.26) must be identically zero for all values of

&

u, h v and

6

w. Therefore, there is obtained from the integrand of the double integral

J N:?C + _d_N_x~y _ LJ h"

a

x

a

y - ,-a u

(2.27)

These are the plate equations of mot ion for moderately large amplitudes. The terms on the right-hand side of each equation in (2.27), namely,

p

hü ,phv and phw are the inert ia terms in the x, y and z direction, respectively. The terms

_f

~..L..! 3 2·' and L) h 3 ",2w (J

12 Jx2 ' 12

Jr

are the rotary inert ia terms. For a detailed discus sion on these terms, see a paper by Mindlin [18J.

l+V dW + -2 Olx E (2.28) 12 Ps(l- »2) {.. . . .;w .. f7 w + = - w + u + v -h2E dX (jy

The boundary conditions associated with large deflections may be

derived from the integrand of the line integral in Eq. (2.26). Formulation

of these conditions for cases of particular interest in aircraft design is

E. Formulation of Boundary Conditions

The governing differential equations of motion given by (2.28) are second order in u and v and fourth order in w. Four boundary condi-tions must therefore be specified at each edge of the plate.

Aircraft skin panels are generally riveted or glued at the edges to supporting structural camponents, such as stringers, ribs, bulkheads, etc. These components are usually rigid enough in a plane perpendicular to the plane of the panel to warrant the assumption that the transverse deflection w is zero along the edges. The torsional stiffness of the supporting com-ponents, however, is, in most cases, insufficient to simulate a clamped edge condition. Even when the attachment between the plate and the supports is adequate to transfer moments, a clamped edge condition would therefore not be obtained. It is thus expected that in practice the actual condition lies somewhere between the simply supported and clamped edge configuratian. This suggests that each edge condition be investigated separately.

If the panel is surrounded by identical rectangular panels which form an array extending in tbe x and y directions, it is plausible to assume that the motian during flutter will have a spatial periodicity in the x and y directions. Consequently, adjacent panels have similar deflec-tions, which could be symmetrie or anti-symmetrie with respect to the cammon edge. Here it is proposed to treat only the anti-symmetrie case because lower resonance frequencies are involved, partieularly when some energy trans-fer exists between adjacent panels.

In this study the behavior of skin panels on the tail surfaces is of particular interest. Consider, for instanee, the array of panels near the center of the vertieal tail. When the deflection w is anti-symmetrie with respect to the common edge between adjacent panels, the deflections u and v, whieh are only significant at large deflection, are (apart from a constant deflectian) symmetrie with respect to the panel edges x

=

ma and y

=

nb ,respeetively. Thus, the edges in the chordwise and spanwise direc-tion remain straight. Beeause the span is large, the condidirec-tion that the edges in the chordwise direction remain straight holds. The eondition that the edges in the spanwise direction remain straight is approximately satis-fied for the spanwise edges in the middle of the array (Fig. 1). This

con-dition, however, does not hold at the edges x

=

0 and x

=

~a unless these edges are supported by a thicker skin which is free of flutter. Assuming that this is the case, u and v become functions of time only along the edges perpendicular to the x- and y-direction, respectively.

To avoid unnecessary complication in the mathematical treatment of the membrane behavior of the plate, it is further assumed that the shear

application of aircraft panels with stringers along the edges. It suffices here because the mechanism of flutter instability is of principal interest and a change of this boundary condition from free movement to zero movement along the edge has a relatively small effect on the displacement w [291.

The analytical expressions for these boundary conditions are form-ulated in the following discussion for rectangular plates with edge lengths a and b. The plate is referred to a Cartesian coordinate system OXyz, the xy-plane being the middle plane of the plate and the origin 0 at a corner of the plate.

1. Simply supported edges: The boundary conditions for this case may be formulated as follows.

For the edges parallel to the y-axis, x

=

0 and x

=

a ,

w = 0 thus

Also, let u = 0 at x = 0 and let u = cl(t) at CJv Jx x

=

0 and

=

0 • x = a • Because at x

=

a ; thus ,;V +

.2.:!!

dW

=

';x Jx,;y ~= 0 (}y

o ,

and so

Returning to the equality (2.26), and realizing that the small quantities Su,

Sv

and Sw shouJ.d also satisfy the boundary condition, there follows from the integrand of the line integral that the coefficient of the term iJ

S

w must be zero

dX

Mx =

0 and hence

(a}x

w_·

_-f

_{0 at x}

₌

_{0, a ) Thus,} or + - = 0

A

dy2

a'2v

- = 0

Jx2

Summari zing, at x

=

0 u = - = w = - = O (Jv

J2v,

ax

tJ-x2 at x

=

a u = cl(t) , -,)v

=

w

= - -

~2w = 0 dX ~X?-(2.29)

where cl(t) is the displacement in the plane of the panel of the edge x = a . Similarly,

at y

=

0

at y

=

b (2.30)

where the displacement in the plane of the panel, v ) is tA.ken eg.ual to zero at the edge y ::: 0 and eq:ual to c2 (t) B.t the edge y

=

b •

2. E:l~mped edges: In an analogous manner the boundary conditions for clamped edges are

at x = 0 u = ~

=

w = dW = 0

ax

Jx

at x

= a

u

=

cl(t) ,

~v

=

w

=

~

= 0 dX tJx (2.31) at y

=

0 v

=

~ = w = ~w = 0 c1y c1y at y

=

b v

=

c2 (t) , .i_,~

= w

=

r) w

=

0 Jy

Jy

(2.32 )

where cl. (t ) of the edges

and ~ (t) are the displacements in the pl.ane of the panel x

=

a and y

=

b , respectively.

lIl. AERODYNAMIC FORCES

A. Unsteady Supersonic Aerodynamic Theory

The fundamental equations which govern the unsteady supersonic isentropic flow past thin, nearly planar bödies have been derived by Garrick and Rubinow [l9J and Evvard

[2Q.

For convenience, these fundamental equations are reproduced here with a brief outline of the derivation for flat or

slightly curved plates. The plate is subjected to supersonic flow on the upper surface only. In addition, the static aerodynamic force resulting from the pressure of undisturbed supersonic flow is neglected. Thus, there

is no statie pressure difference between the upper and lower surfaces. If the lower surface of the plate is subjected to still air, pressure fluetua-tions also result on this surfaee due to the motion of the plate. These forces are also neglected to avoid unnecessary complication.

The linearized partial differential equation for the perturbat10n velocity potential ~ in steady or unsteady supersonie flow takes the form

2M

(3.1)

=

where M is the free stream Maeh number andcoo is the velocity of sound. The flow is steady and uniform at infinity and in the direct ion of the x-axis. The transformation

x "'x

y

=

~1_M2

Y

=

~1_M2

(3.2)

z z

which is a combination of the Lorentz and Galilean transformation, converts

E~. (3.l) into

(3.3)

Fundamental solutions of (3.3),corresponding to spherical waves from which general solutions may be formed, are

and where

1>b

= !....f

_R* R*

=~x

2 +

y

2 + Z 2

-

~)J

coo

f

=

arbitrary function representing a source strength (3.4)

(3.5)

(3.6)

In (3.4) and (3~5) the fixed source is located at the origin of the

x ,

y ,

z space. The spherical waves are converging onto the source in (3.4) and are diverging from the center of the disturbance in (3.5). Transforming

t

back to t by

(3.2)

there is obtained from (3.4) and (3.5)

~a

=

1

R*

fa [t Mx+R*

J

f:J 2cco

(3.7) and

~b

=

~*

fb [t Mx-R*

]

fJ

2_{c co} (3.8)

It is recognized that the solutions (3.4), (3.5) and (3.7), (3.8) are on1y va1id "in the" downstream Mach cone whose apex is at thè disturbance source

(here the origin).

Introducing the disturbance at the point

(ë"

1? '

~) instead of at the origin, (3.7) and (3.8) become

=!

f

Ct -

M( x- ~) -I-R )

=

!

f R f12c R a 00 (3.9) and (h ( _{~b x,y,z,t} )

_{= R-}

1 _f

Ct _

M(x- '@,) -R) =

1:

fb

f3

2 coo R (3.10) where R

=~(

x ..

~)

2 +

ei ..

?f)

2 + (Z" _

~)

2

=~(x- ~)2

_ /32 [(y_"'l)2 + (z-

~

)2] (see (3.2»

Garrick and Rubinow have shown [19] that the appropriate solution is the sum of Eqs. (3.9) and (3.10).

For disturbances distributed in space, (3.9) and (3.10) are re-garded as e1ementary solutions corresponding to a disturbance source of strength f

=

fd ~ d ~ d -; located at the point ( ~ ,

'l '

~

)

"

.

'!'he tota1 solution is then obtained by superposition of e1ementary solutions, justi-fied for 1inear equations. '!'his is accomp1ished by integrating with respect

(3.ll)

The region of integration S is in supersonic flow restricted to the forward Mach cone with apex at the point (x,y,z) .

B. Equations for Nearly Flat Plates

For flat or slightly curved plates, in the plane of the plate defined by ~

=

0 • sources (3.11) can therefore be replaced by a

Z;

=

0 and (3.11) changes into

the sources of disturbance are The volume distribution of surface distribution over

(3.12 )

The strengths fa and tb of the sources have to be determined so that the condition of tangential flow on the surface of the plate is satisfied. In linearized theory this boundary condition is satisfied on the plane z

=

0 instead of at the surface itself. For a point in the plane z

=

0 the area of integration S in (3.12) is the upstream Mach cone of (x,y,z) defined by the lines

"7

=

y :t

1

(x-ë,) j (x-

~

)

>

0 •

Since the plate is subjected to supersonic flow over the upper surface only, the boundary condition on the z = 0 plane becames

(3.13 )

where W(x,y,t) is the local z component of the perturbation velocity of the upper surface measured positive outward of the plate.

It can be shown from (3.12) (see [19J, Appendix C) that

lim

z~O+

~~(x,y,

z, t)

=

-2 -n-f(x,y, t)

It is convenient to introduce the following notation:

(3.15)

where '"ra and "t'b are interpreted as time delays •

.

In this notation, there is obtained from (3.9) and (3.10), remem-bering that ~

=

0

cp

_a (x,y,z,t) = f(t- "t'a)

~(x_~)2_,B2

[(y_"7)2+z2] and

~b(x,y,z,t)

= f (t- "t'b)

~(x_~)2_

13

2 _{[(y-7Z )2+z2]} From (3.13) and (3.14)

r(t- 1:'a)

= -

~ W(x,y,t- "t'a)

27t and (3.16) (3.17) (3.18) (3.19)

The velocity potentia1 in the upper half space may therefore be written as (see (3.12»

1>

(x,y,z,t) u

w

(

~

,

~, t - 't' a) + W (

ë.

,'>2,

t -~ b)

~

(x-

i

)2",;92 [(Y-1(

)~\z2

]

(3.20)

The potential on the upper p1ate surface, where z

=

0+, thus becomes

~u(x,y,o+,t) (3.21)

The perturbation pressure at any point (x,y) ( an overpressure (p-poo >0) is counted positive) arising from the motion of the plate on1y may be written in terms of the velocity potentia1 as

(3.22)

where ~

=

the air density in undisturbed supersonic flow

u

=

forward velocity

The vertical disturbanee velocity at the point (x,y) , W{x,y,t) , is

dw

dt"=

w{x,y,t) ::: (3.23)

For the purpose of flutter ana1ysis, the aerodynamic force per unit area in transverse direction of the p1ate is thus given in terms of w by (3.22), (3.21) and (3.23). These equations are va1id provided that 1inearization of the exact, non1inear, unsteady flow equation is justified. The 1inearization procedure is based upon the assumption that the slopes

aw and ~ ,remain sma11 compared to unity, which is equivalent to saying

01 x rly

holds true well into the region where the nonlinear elastic behavior of thin plates must be considered because, in the latter case, the criterion is that w be larger than the plate thickness. Linearized aerodynamic theory may therefore be applied in the derivation of the flutter equations of motion for large amplitude vibrations of thin plates.

The assumption is now introduced that the plate has an infinite span and is separated into rectangular panels, each with lengths a and b in x and y directions, respectively (see Fig. 1). In addition, the transverse deflection is continuous at the boundaries of each panel and is anti-symmetrie with respect to the common edge between adjacent panels. With this hypothesis, it is not necessary to account for the boundaries of each panel in the limits of integration of the expression for the velocity potential. The S region can thus be defined as that area of the x-y plane within the forward Mach lines

." == y ±

!...

(x- ~) ; (x- ~)

>

0

p

(3.24 )

and the leading edge. The trailing edge of the plate is at

x ==

.L

a (3.25)

where

i

equals positive integer. With these boundaries, the velocity potential takes the form

<Pu(x,y,t)

=

1 27r

Let the time-wise schedule for the Uvwash be harmonic.* The

upwash is the imaginary part of the complex variable

(3.2?)

Thus, using

(3.15)

with z

=

0 ,

so that

jwt

e

x y+~(x-ê,)

I /

f3

W(

ë: , "'(

)e-j

_U3(x-

e;)~

_cos(ü5

_R)

d7(d~

1 R M

o

y- -

(x-

t:)

=

-,B (3.29) where (3.30)

The variable

"'7

is replaced for fixed y, x and ~ by a variable

e

with the transformation

'?

=

Y - ~ (x- e;) cos 9

(3.31)

It readily follows that (3.29) becomes

<p (x,y,t) u

=

-ejwt

j'X

11t'-

{1

}

-jÜ>(x-

~)

O W 1E"y-

-;9

(x- ê; )eos Q e

7r1 ,

0

x

cos { : (x-

Ei

)sin

Q}

dGd~

C. Solution of Potential alld Pressure Equations for an Array of Rectangular Panels with Simply Supported Edges

(3.32)

Luke and St. John

[7]

applied the theory described in the previous ehapter to the case of an infinite span panel separated into rectangular bays by equally spaeed rigid stiffeners in the streamwise direction. In this

chapter the work of Luke and St. John is extended to the case of an array of panels (see Fig.

1).

Now suppose that the geometrie part of the upwash is given in separable form as

(3.33)

If the chordwise edges (y=O,b) are simply supported, the transverse deflec-tion in the spanwise direcdeflec-tion ean be-described as

= sin n"l( "'?

b (3.34)

Designating the eontribution of rn(~)gn(~) to the total potêntial as

~u,n'

(3.32)-(3.34) yield

.. '\ X 7t

,+. ejU/t

11

[n1t'y (n1'C ) 'f (x,y,t)= - Fn( ~) sin - - cos - (x-~ )cos 9 +

u,n

7(/1

.

b bfi

o

0

But

(3.36)

since the integrand is odd ab out 9

=

~/2

that

Further, it is deduced from [21]

(3.37)

where Jo is the Besse1 function of the first kind of order zero. Thus (3.35) reduces to

1>

_u,n (x,y,t)

=

where x sin n1'ty

r

F

(~)G(x-~

)dë, b

j o n

(3.38) (3.39)

It is noted that in (3.38) the spanwise mode shape sin

n~y

appears as a simp1e factor outside the integral of the potential function. This means that there is no aerodynamic coupling between spanwise modes. In the aerodynamic terms each spanwise mode ean thus be treated separately.

Let the transverse deflection w in accordanee with (3.27) and (3.33) be given by

From (3.23), (3.27), (3.33) and (3.40) Substitution of (3.41) in (3.38) gives Now, and sa toot x jwt

1

~ (x,y,t)= _ e U sin ~ u,n fJ b

_o

(3.41 ) (3.42) x

;:

(3.43)

The expression for the perturbation pressure is obtained by coupling (3.43) and (3.22) and performing the necessary operations. Thus,

P (x,y,t)= eu2ejwt sin n-rr:y [{Jfn(X) + 2 j(Û

~(x)}

G(O) +

u,n f7 b Jx U dG(

~)

I

+ a~ at~ =x (3.44) _ ju) U fn(O) G(x) +

where Pu n is the contribution of gn(-l() (see (3.34» to the total per-turbation'pressure. Now JG(

~)

I -

ju) , G(O)

=

1 J ë, at& =0 -(3.45) and also fn(O) ::; 0 (3.46)

Utilizing (3.39 ), (3.45 ), (3.46) and the relationsbips

d Jo(x)

=

- Jl(x) Jx J _{2Jo (x)} 1 (3.47) = - Jo(x) +

x

Jl(x)

Jx2

(3.48)

At this stage introduce dimensionless variables Xl, y', etc., by writing

x

=

axl

,

_y

=

_ay'

(3.49)

~

=

a ~ , and f(x}

=

af' (x' ) Also define k=~ ; s=! U b kM -

~

2 (n'1t' s )2 K

=

(32;

r=ar

=

K +

f

(3.50)

P~

_,

n(x'

,y'

,t)

=

jwt e ,6 sin n 7t sy'

x

[

~.::...-

~f~(X') + jk(M 2-22 )fJ(xt) +

a

x'

f1

(3.51)

It is interesting to note that the terms not contained in the inte-gral correspond to the approximation used in quasi-steady aerodynamic theory

[3

J.

In the "quasi-steady" approximation terms up to the first power of k are only retained in the power series expansion of the three-dimensional solu-tion for the velocity potential. As indicated by Shen

[3],

the quasi-steadY expression furnishes a good approximation if the reduced frequency k is small and the Mach number is sufficiently removed from .-.,J2.

Following the work of Luke [22J, the Bessel functions can be ap-proximated by a sum of circular functions as follows.

q

=

!.

L

[Sin

2 (2r-l)

"'7'(J

cos(

~r

ê, I) +

q

r=l

4q

(3.52)

where

(3.55 )

In these approximations the term(s) appearing outside of the sum-mation sign give an estimate of the error in the representation and are con-sequently omitted in the following ana1.ysis. These approximations are very powerful.. For example, if q =:3 and r~' ~5.0 , at least three decimal accuracy is assured. The determination of the accuracy needed for satisfac-tory results in flutter analysis is, however, difficult. The numeri cal. cal-culations are therefore carried out for different values of q and on this basis the value of q needed was decided. It was found that for the numer-ieal results presented in Section VI sufficiently accurate results can be obtained with q

=

3 •

Tbe approximation of the Bessel functions by eircular functions is chosen to simp1ify the eva1uation of the integral in (3.51). In the next stages of the anal.ysis, this point wi11 become evident.

Let 1 [ 2....2 2 (2r-1) ] ar

= -

q

(KM-k) + I' cos

4q

7t' and (3.56) = 2rk cos

C

2r -1 ) 7( qf32 4q

Substitution of (3.52)-(3.55) in (3.51) and using (3.56) yields

p~

n(x',y',t)

=

e jwt sin n7rsy'

[tlf'

n

(x')

+ jk(M -2)f '2 _{(X ' )} +

,

!1

d x' ;92 n

Of particular interest in this report is the case where the chord-wise edges of the panel are also simply supported. Therefore, let

CP

f~(x') =

L

q~,n sin m'1!"x' m=l

(3.58)

where q~,n is the nondimensional generalized coordinate in transverse direction. The second index on q~,n is introduced to signify the associ-ation of the coordinate with the spanwise mode shape sin n"7t'sy' •

and

Substituting

(3.58)

into

(3.57)

and utilizing

J

_e ax' _{sin(bx'+c)dx'}

J

ax' , e cos(bx'+c)dx'

=

eax' {a sin(bx'+c)-b cOS(bx'+c)} a2+b2 = eax' {a cos(bx'+c)+b Sin(bx'+c)} a2+b2

there is obtained af ter some algebraic manipulations

p~

_,

n (x'

,y' ,

t)

=

where jw t e - sin n'l'Csy'

fJ

(3.59)

(3.60.)

(3.62)

(3.63)

q F _m =

L

_{(a.rer ,m+brc'r ,m)}

(3.64)

r=l q ~ =

L

_{(a.rdr,m+brfr,m)}

(3.65)

r=l

and p' (x',y',t) signifies the a.erodynamic nondimensionalized perturbation

u,n

pressure due to a transverse deflection

00

W' (Xl

,y'

,t)

=

eiwt

2:

~

sinm?rx' sinn7t'sy

m=l ,n

For the solution of the flutter equations of motion, the following expression is needed

L

I _ ejwt

=

1

2s n,m

I

1

i

s

11

p~ n(x;y;t) sin m7r"x' sin n7r"sy' dy'dx'

0 0 '

and

that

Combining (3.59) and (3.66) one deduces with

i.

la

sinm7rx' sinm7<x'd.X' =0 if m=f=m m=m

1

cos m"'x' sin JJJ7tx'dx' = 0 U (m+iii) l i s even

o

I _n,m -

=

1

fS

2

J,..,

sin nTtsy'dy'

o

1

= -

_2s + -1 { (m7r -F )

€ - -

- (-1)

mi

e-J

°KMR.-

_{G - +}

1

m m,m m,m i f (m+ffi)

i

is odd

~,m}

]

in which the fo11owing definitions pertain.

$ _ = 0 if m =1=

m

m,m

=

1 if m

=

m

-E _ = 0 i f (m +

m)i

is even m,m (3.67) (3.68) (3.69) (3.70) (3.71) (3.72)

and ~ = ;[

{(~r

m

m+b~r

m iii)cOS "r

i

+ r=l ' , , , +

j(a.~r,m,m+brSr,m,m)sin

"'r

i }

ft -

=

± {

(a g -+b h _) } Jm,m r=l r r,m,m r r,m,m g _r,m,m

-IV. SIMPLIFICATION OF LARGE AMPLITUDE FLUTrER EQUATIONS

FOR AN ARRAY OF RECTANGUI.l$ PLATES

A. Initial Simplifications of Equations

(3.73)

(3.74)

(3.75)

(3.76)

The flutter equations of motion for large amplitude vibrations of plates subjected to supersonic flow over the upper surface only may be obtained from the previous sections by substituting the integra.l relations for the aerodyna.mic forces, Pu (see (3.22), (3.21) end (3.23»,into the set of nonlinear partial differentia.l equations (2.28). Formal mathematical treatment of these equations subject to appropriate boundary conditions for rectangular plates, such as (2.29), (2.30) or (2.31), (2.32), is extremely complex a.nd involved if at all possible. The ma.in difficulty is, of course, the nonlinear terms appearing in (2.28). It is therefore essential that the equa.tions be simplified.

As a step in this direction, it is noted that the first two equa-tions of (2.28) describe the motion (campression and tension) in the plane of the plate. The natural frequencies of vibration in this plane are re-lated to the velocity of wave propagation of the plate material and are, in general, quite high and certainly much higher than the frequencies of the lower modes of vibration in the transverse direction. In addition, no direct excitation of in-plane vibrations is obtained by the aerodynamic forces and in-plane motions occur only due to nonlinear coupling with trans-verse motions. In panel flutter investigations, main interest is concen-trated on the lower modes in transverse direct ion, and the effects caused by the propagation of disturbances in the plane of the plate may therefore

fJ

_s(1-V 2 ) ••

- - - u and be neglected. This implies that the inertia terms

E

Ps

(1- ))2) ••

- - - v may be omitted in the first and second equation of (2.28), E

respectively. Consistent application of this approximation to the third

12 f's(1-V2) ..

aw

d

equation of (2.28) means that the terms u - an

h2E i?x

12

P

s (1-V 2) .. J w

- - - v - must also be omitted.

h~

ay

For another simplification note that the rotary inertia terms in the third equation,

small compared to the inertia term in transverse deflection

, are usually

12 l's (l-)} 2)

h2E

..

w •

The ratio of the rotary inertia terms to the transverse inertia terms is in the order of plate th:'cl{.!J.er;;s sguar'ed. divicl€d by wavelength squared, so that the rotary inertia t~:rrrw c<.n aJ.SO be neeJ.ected.

Performing th'3 indica'ted simpJ.ifications and rearranging the terms, (2.28) becomes

+ (I-V)

aw aw

J2w ]

Jx ay t7Xay

=

..

w