SP 8004 - Panel Flutter

8/11/2019 SP 8004 - Panel Flutter

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N A S A

S P A C E V E H I C L E

D E S I G N C R I T E R I A

NASA SP-8004

S T R U C T U R E S )

PANEL

FLUTTER

JULY i 964

Rev i sed

JUNE

1972

NATIONAL A E S N

A U

T

I

S

A N

D S P A C E A D M N ST RAT

LI

r\i


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FOREWORD

NASA experience has indicated a need for uniform criter ia for the design of space

vehicles. Accordingly, cr iteria a re being developed in th e following areas of techno logy:

Environment

Structures

Guidance and Control

Chemical Propulsion

Individual components of this work will be issued as separate monographs as soon as

they are completed. A list of all published monographs in this ser ies can be found at

the end of th i s docum ent .

These monographs are to be regarded as

guides

to the formulat ion of des ign

requirem ents and sp ecif ications by NASA Centers and p roject off ices.

This monograph was prepared under the cognizance of the Langley Research Center .

The Task Manager was

G .

W. Jones , J r . The author was E. H. Dowel1

of

Princeton

University. A number of other individuals assisted in developing the material and

reviewing the drafts . In particular , the significant co ntr ibu tio ns ma de by the following

are hereby acknowledged: C. P . Berry, D. L. Keeton, and D. A. S tewar t

of

McDonnell

Douglas Corporat ion; J . Dugundji of Massachusetts Institute of Technology; L. D. Guy

of

NASA Langley R esearch C enter; M. H . Lock of Th e Aerospace C orpo rat io n; M. H.

Shirk of U.S. Air Force Fl ight Dynamics Labo ratory; and H . M. Voss of Boeing.

NASA plans

to

upd ate this mono graph periodically as appropr ia te . Com men ts and

recomm ended changes in t he technical content are invi ted an d should be forwarded

to

the a t ten t ion of the Structural Systems Office , Langley Research Center , Ham pton ,

Virginia

23365.

J u n e 1972


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GUIDE TO THE

USE

OF THIS MONOGRAPH

The

purpo se of this monogra ph is to provide a uniform basis for design of flightworthy

stru ctu re. It sum marizes fo r use in space vehicie deveiopiiieiit :he significant experien ce

and knowledge accumulated in research, development, and operational programs to

date. It can be used to improve consistency in design, efficiency of the design effort,

and confidence in the s tructure. All monographs in this series employ the same basic

fo rma t

--

three major sections preceded by a brief INTRODUCTION, Section

1

and

complemented by a l is t of REFERENCES.

The STATE

OF

THE A RT , Sect ion

2,

reviews and assesses curr en t design p ractices a nd

identifies important aspects of the present state of technology. Selected references are

cited to supply supporting information. This section serves as a survey of the subject

th at provides backgroun d m aterial and prepares a proper technological base for th e

CRIT ERIA and RECOMMENDED PRACTICES.

The C RITER IA, Sec t ion 3 , s ta te

wh t

rules, guides, or limitations must be imposed to

ensure flightworthiness. The criteria can serve as a checklist for guiding a design or

assessing its ade qu acy .

The RECOMMENDED PRACTICES, Sec t ion 4, s ta te

how

to satisfy the cri teria .

Whenever possible, the best procedure is described; when this cannot be done,

app ropr iate references are suggested. These practices , in co njun ction w ith th e criteria ,

provide guidance to the formulation of requirem ents for vehicle design and evaluation .


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CONTENTS

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

STATE OF THE ART 3

2.1 Consideration of Flu tter in Panel Design

. . . . . . . . . . . . . . . .

4

2.1.1 Flutte r-Re sistan t Design

. . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Flu tter Margins and Conservative Assu mption s . . . . . . . . .

5

2.1.3 Panel Flu tter Prediction in Preliminary Design . . . . . . . . .

6

2.2.1 Structural Parameters . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Aerod ynam ic Parameters

. . . . . . . . . . . . . . . . . . . . 10

2.2.3 Assessment

of

Panel Flutter Theory . . . . . . . . . . . . . . . 1

2.3 Panel Flut te r Tests

. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2.3.1 W ind-Tunnel Panel-F lutter Testing . . . . . . . . . . . . . . . 13

2.3.2 Flight Flut ter Test ing

. . . . . . . . . . . . . . . . . . . . . .

15

2.4 Correlat ion of Analytical and Test Results . . . . . . . . . . . . . . . 15

2.2 Panel Flu tter Analysis

. . . . . . . . . . . . . . . . . . . . . . . . . .

7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

.

CRITERIA 19

3.1 Analyses and Model Tests . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Nondestruct ive, Limited-Ampli tude Flut ter . . . . . . . . . . . . . . . 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 2 Flight Tests 20

4 RECOMMENDED PRACTlCES . . . . . . . . . . . . . . . . . . . . . . . 21

4.1

Analyses and Model Tests . . . . . . . . . . . . . . . . . . . . . . . . 2 2

4.1.1 Structural Parameters

. . . . . . . . . . . . . . . . . . . . . . .

2 3

4.1.2 Aerod ynam ic Parameters . . . . . . . . . . . . . . . . . . . . . 2 4

4.2 Fl ightTests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Nondestruct ive, Limited-Ampli tude Flut ter . . . . . . . . . . . . . . . 25

APPENDIX Imp ortant Structural and Aerodynamic Parameters

. . . . . . . 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

REFERENCES 3 7

V


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NA SA SPACE VEHICLE DESIGN CRITERlA

MONOGRAPHS ISSUE D TO DATE . . . . . . . .

.

. . . . .

.

.

.

. .

.

. 45

v i


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PANEL FLUTTER

1

INTRODUCTION

Panel fl utte r is a self-excited, dynamic-aero elastic instability

of

thin plate or shell-like

componen t s of

a

vehicle. t

O C C L ~

ost frequently, though not exclusively, in a

supersonic f low. At subsonic speeds, the instabil i ty more often takes the form of a

static divergence or aeroelastic buckling. Flutter is caused and maintained by an

interact ion amon g the aerodynam ic, inert ial , and elastic forces of the system. Init ially,

the ampli tude of the mo tion of an unstable panel increases expon ential ly with t im e,

al though frequently the ampli tude

is

limited because

of

nonlinearities, usually

structural .

Panels are normally designed to avoid flutter. If it should occur during flight, however,

then l imited-ampli tude and l imited-durat ion flut ter may be tolerated for some vehicles

as long as the ampli tude and durat ion do not cause:

( 1 )

structural fai lure of th e panel

or support ing stru cture d ue to fat igue, (2) functional fai lure

of

equipment a t tached to

the s t ruc ture , or (3) excessive noise levels in space vehicle com par tm ent s near th e

flutterin g panel.

Panel flutter has occurred on a number of flight vehicles. Early experience, largely

aircraft , is surveyed in reference 1. More recently, panel f lut ter has occurred on the

X-15 during fl ight operat ion (ref.

2 ,

during wind tunnel tests in the development

program of the X-20 (refs. 3 t o 5), on Titan I1 and I11 (ref.

6),

and on the S-IVB

(ref.

7).

The structural damage resulting from panel f lut ter was judged destructive on th e X-15,

the X-20, and the aircraft . The structure of these vehicles was sti ffened t o prevent

panel f lut ter throu gho ut the fl ight envelope. For the Titans and S-IVB, the flut ter was

judged nondestruct ive because i t was determined that the severi ty and durat ion of the

flut ter would not be great enough to degrade unacceptably the structural integri ty of

the panel . H ence, no st i ffening was added (and no weight penalty incu rred) to prevent

flut ter of these panels.

This monograph is concerned with the predict ion of panel f lut ter , determination of i ts

occurrence, design for i ts prevention, and evaluation of i ts severi ty. Theoret ical

analyses recommended for the predict ion

of

flutter stability boundaries, vibration

ampli tudes, and frequencies for several types of panels are described. Vibration tests

and wind tunnel tests are recommended for certain panels and environmental f low

condit ions to provide info rmation for design or verif icat ion of analysis. App ropriate


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design margins on flutter stability boundaries are given and general criteria are

presented for evaluating the severity of possible short-duration, l imited-amplitude

panel f lutter on non-reusable vehicles.

Th e occurrence of f lutt er in a particular panel configuration depend s upon the mass,

damping, and stiffness of the panel; local Mach number, dynamic pressure, density;

in-plane f low a ngular i ty; and, fo r some condi tions , boundary layer profi le a nd

thickness. Th e para meters affecting panel stif fness w hich are ref lected in panel natural

frequencies include the panel length, thickness, m aterial modu lus, length-to-wid th

ratio, edge conditions, curvature, or thotropy (variation in stiffness with direction) ,

in-plane loads, transverse pressure differential across the panel, and acoustic cavity

(closed-in space) beneath the panel. For

some

configurations geometr ic imperfections

in the panel may be i mp ort an t as well .

Related NAS A design cr iter ia monographs include those on natural vibration modal

analysis (ref.

8);

struc tural vibration prediction (ref .

9);

and f lut ter , buzz, and

divergence of lifting surfaces (ref.

10).

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2 .

STATE OF THE ART

On e of the diff icul ties in assessing the state of the art with respect t o panel f lut ter is

the large num be r of parameters which may be important for any part icular app lication.

References 1 1 and 12 present some

of

the historical background of the problem . More

recent surveys are those of references 13 and 14,which give bibliographies c om plete to

the t im e of pub lication. M uch of th is state-of-the-art section is based o n reference 15 ,

deal ing w ith theoret ical aspects of panel f lut ter , and reference 16 , which is concerned

primari ly with the ex perimeniai aspects and theoretical-experimental correlation of t h e

problem . T wo addit ional general references tha t are of great use are reference 17 ,

which is a survey

of

the l i terature on free vibrat ions of plates, and reference 18, which

gives simplified criteria in graphical form

for

mo st, though not al l , of those param eters

which may be important for panel f lut ter design. Lit t le previous knowledge of the

subject is assumed on the part of the reader of reference 18,and the premise is that n o

panel

shall be permit ted t o experience f lu t ter; however, reference 18 provides no

method for the inclus ion of boundary layer ef fects , for handl ing or thot ropy and

damping, o r fo r handling pressurized o r buckled panels accurately.

Some background knowledge of the physical nature of the panel-flutter problem is

useful for assessing the state of the art . The f lut ter boundary is commonly defined as

the variat ion with Mach num ber of the d ynam ic pressure at which the on set of panel

flutter begins (refs. 19 t o 22) . Below the f lut te r boundary, rand om osci l lat ions of the

panel occur which have predominant frequency components near the panels lower

natu ral frequenc ies. These oscillations are th e panel response to pressure fluctu ation s in

the turbulent bound ary layer ( i .e. , noise) . The amp li tudes of the oscillations are

normally some small fract ion

of

the panel thickness.

A s

the f lut ter boundary is

exceeded at some cri t ical dynam ic pressure, cal led the f lut ter dyn am ic pressure, qp the

oscillation becomes nearly sinusoidal with a n ampli tude that tends to increase with the

dynam ic pressure and approach es or exceeds the plate thickness. On e major limitation

of the present state of the art is the lack of data covering an extensive range of

dynam ic pressure, q, greater than

qf

particularly at supersonic speeds. However, limited

dat a of thi s type have been obta ined for

S-IVB

type panels (ref. 23).

F lut ter onset i s more a matter of defini t ion than i t is some point which can be

determined with great precision. Using the best available techniques, the onset can be

es t imated wi thin about

10

percent of t he dynamic pressure (ref . 21). There has been

some ef for t to obta in a more precise experimental determination

of

the f lut ter

bound ary by using ad mi t tan ce techniques and the concept of a linear plate impedance

(ref . 24) .

The behavior of panels aft er flut ter onset is largely domin ated by s ystem non linearities,

t he mos t p rominen t

of

which is the nonlinear structural coupling between bending and

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stretching of the plate. The plate s tretches as i t bends, thereby inducing a tension in

the plate. Limited-amplitude, post-flutter onset oscillation results from a balance

between the (unstable) l inear-plate and fluid forces and this tension force, which

increases the effective plate stiffness. Qualitative estimates of the flutter amplitude that

account for this balance can be made by order-of-magnitude considerations (ref.

25).

Not a great deal of s tudy has been directed toward flutter fai lure mechanisms;

however, at least two are readily identifiable and have occurred in practice. If the

flutter-induced stress level exceeds the yield stress of the plate material, then

catastrophic or rapid failure occurs; on t he o th er han d, even a relatively low stress level

s temming from a sustained period of flutter can induce fatigue or long-term failure.

Fatigue life can be estima ted if the stress level and frequ ency of the oscillation are

known. Current analytical methods are inadequate for predicting failure mechanisms;

hence, wind tunnel or fl ight flutter tests must be co ndu cted f or this purpose.

Current practice is to design a panel to avoid any flutter . However, should flutter o ccur

during developmental test ing or fl ight operation, the designer has, on occasion,

exercised the option of demonstrating that f lutter is nondestructive rather than

redesigning the panel. This approach is normally only attempted for short-l ived,

nonreusable op eratio nal vehicles.

2.1Considerat ion

o f

Flutter in Panel Design

Conventional practice in the initial structural design of panels has been to design each

panel to withstand the s teady and dynamic load environm ents i t is expected to

encounter with little or no consideration given to a possible panel-flutter instability.

However, certain rules-of-thumb have been developed which lead to increased

resistance to panel flutter w ithou t the necessity of detailed analysis o r testing.

2.1.1 Flut ter-Resistant Design

Minimum-gage panels are particularly flutter-prone. Conversely, panels designed to

withstand large static (e.g., compressive, lateral) or dynamic (e.g., acoustic, bending)

loads are apt to be so thick that f lutter is not l ikely to occur. Because of the many

possible panel co nfiguration s, general guidelines to flutter-resistant pa nel design have

not been well documen ted in the l i terature. Nevertheless , the following guidelines for

flutter-resistant design have em erged:

Align sho rt edges of rectan gular p anels parallel to the airflow , and stiffeners

in stiffened panels also parallel to the flow, and where feasible, provide extra

stiffening of edge sup ports perpendicular to the panel s t iffeners .

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e

e

e

e

Avoid designs having closely spaced natural frequencies, or natural fre-

quencies which are abn ormally sensi tive to any parameter.

Where panel configurations cause flutter behavior that is sensitive to

structural damping or geometrical imperfections, make design changes to

eliminate this sensitivity. (Normally, such changes involve the separation

of

closely spaced natural frequencies.)

rariei

ciirvature perper?dici.iIar to the direction of airflow is beneficial, but

curvature in the same direct ion as the flow is to be avoided.

In o rde r to avoid destabilizing loads, design panels fo r compressive loads to

have the loading in th e spanwise rather than streamw ise directio n.

2 1 2 Flu t t e r Marg ins and Con servat ive Assumpt ions

General ly speaking, the abil ity to predict panel f lut ter by experim ental and theoret ical

means has improved greatly in the past ten years. However, there are still panel

configurat ions, loadings, and flow condit ions for which the understanding of and

ability to predict panel flutter are lacking. Hence, the current practice is to use

conservative assumptions for panel or f low parameters to ensure an adequate panel

design. In add it ion, a m argin on flut ter dy namic pressure is often specified t o al low for

the uncertainty in some instances as to w hat co nst i tutes a conservative assumption

(i .e . , an assumption which leads to the predict ion

of

a lower f lut ter dy nam ic pressure

than that encountered in pract ice). By tradit ion, and also on the basis

of

the

differences observed between the results o f theory and ex perim ent, a margin

of

50 percent o n flut ter dynam ic pressure is frequently used.

An overly conservative assumption or several moderately conservative assumptions

which have a cu mulative eff ect, may result in an excessively thic k (hen ce, heavy)

structure. The designer has several alternatives to avoid an excessive weight penalty.

Firs t, he may make basic ch anges in the panel design tha t will result in flutte r

resistance with no weight penalty. This is usually impossible because conventional

practice is to design th e basic str uctur al con figuration initially on th e basis

of

o the r

load condit ions.

Second ly, the designer may use more accurate (but usually more comp licated) meth ods

to est imate the flut ter dynamic pressure and hence reduce the uncertainty and

conservatism in the determination of f lut ter dynamic pressure, due to overly

conservative assumptions. This is frequently done and leads t o a hierarchy of metho ds

ranging fro m theoret ical analyses to wind-tunnel model test ing t o f l ight test ing

of

t he

full-scale st ruc tur e.

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Lastly, i f i t has become clear from fl ight test data that f lut ter does occur, but that i t

may not be damaging, the designer, rather than redesign the vehicle, may at tempt

to

demonstra te tha t such f lu t te r does not compromise the in tegri ty of the vehicle o r i t s

mission. This demon strat ion requires the de termination o f the flu t ter dynam ic pressure

and t he panel ampli tude an d frequency in the flu t ter regime (Le., beyond the flu t ter

boundary),

so

that a fatigue or failure analysis can be made to assess the damage

potential of the flut ter . Such a demonstrat ion has occasionally been made on

short-lived, nonreusable vehicles. The potential damage may take the form of excessive

noise or excessive vibration, as well as structural fatigue. N o generally agreed upon

margins for these typ es o f damage have been developed.

2.1.3 Pane l F lu t te r P red ic t ion in P re l im inary Des ign

Many designers predict p anel flu tter bound aries in preliminary design thro ug h the use

of

design charts based upo n theoretical and experim ental dat a for certain panel

configurations and flow condit ions. In addit ion to reference 18 , which contains such

design charts, special mention should be ma de of references 26 to 28.

Reference 26 c onta ins empirical and theoretical results fo r flat, rectangular panels

under compressive loads in terms of f lut ter dynamic pressure (at high Mach number)

versus panel length/width r at io. Equivalent length/width rat ios for orth otro pic panels

(panels with different but constant st i ffness in two direct ions) are given in terms of

isotropic panels (panels with same stiffness in all directions). Although the limitations

of these results with respect to Mach number and unknown variat ions in test

condit ions are no w well appreciated, this docu me nt continues t o be w idely used.

Reference 27 provides addit ional data of the type presented in reference 26 and also

presents a discussion of the accuracy and usefulness of such d ata.

Design charts are developed in reference 28 for rectangular, isotropic panels (again at

high Mach n um bers ) on th e verge o f buckling (a critical design co nd itio n) using

theoretical methods. Correlations with

a

l imi ted quant i ty of exper imenta l da ta a re

offered to su pp ort the theoret ical results. A l imitat ion o f the theoretical m eth od s is the

necessity of specifying the str uct ura l dam ping of the pan el. Also, cautio n is required in

applying the results of references

18

and 26 to

28

a t low supersonic-transonic Mach

numbers and for pressurized or buckled panels where the simplified nondimensional

correlat ing parame ters used in these references are inadeq uate.

Nevertheless, results such as those given in references

18

and 26 t o 2 8 a re useful fo r

prel iminary evaluation of panel f l ut te r i f on e keeps in mind the l im itat ions of the d ata

and approaches. These sources are freque ntly used

to

ma ke a n initial assessm ent of all

panels in an effort to identify those which require more detai led study. If a

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f lutter-dynamic-pressure margin of 2 or m ore is indicated for some panels, these panels

are of ten considered to pose no ser ious f lut ter problem. The number of dif ferent

possible panel co nfiguration s subject t o varying f low cond itions usually m ake a

comprehensive study of each configuration impossible. Hence some simple screening

me tho d such as tha t just described mu st be used to identify those panels mo st l ikely to

encounter f lut ter so tha t the design effort can be most effectively expen ded .

2 2

Panel

Flut ter Analysis

I t

is

essential to exam ine the structu ral and aerodynam ic parameters systematicaiiy and

assess their relative importance to, and our present abili ty to predict their effect on,

panel f lutter . Parameters in the form er category characterize the mass, stif fness, and

damping of the panel or , alternatively, the modal mass, natural frequencies, and

damping of the structu re; parameters in the latter category describe the natu re of the

flow (e.g. , subsonic or supersonic Mach nu mb er, mass density, and dyn am ic pressure) .

2.2.1 St ructura l Parameters

Th e im portance of the structural parameters for any specif ic panel f lutter analysis can

be assessed by noting their effect o n the panels natural frequencies. Th e abili ty to

determ ine accurately th e effect of these parameters on f lutter can be measured by the

accuracy with which the natural frequencies of the panel can be predicted. The effect

of the structural parameters on the panels natural modes and frequencies can be

determined either theoretically or experimentally. Normally, the most eff icient

procedure is to use theoretical methods to as great extent as possible with occasional

experimental checks to verify the accuracy

of

the theoret ical model . Typical methods

of

analysis used are R ayleigh -Ritz, Ga lerki n, finite-difference, and finite-eleme nt

methods as well as exact solutions to the structural equilibr ium equations (refs. 8 t o

10, 17,

a nd 29 t o 32 ) .

The following structu ral parameters are adequately han dled by classical linear plate or

shell theo ry: plate thickness, mo dulus of elasticity, length, length-to-wid th ratio

( refs . 19, 20, 29, and 30 , acoustic cavity effect (ref . 19 ) , or th otr op y (refs. 26 and

33

to 40) , and, for many cases, f lexural boundary conditions (refs.

1 7 ,

38 , 39 , a nd 41 )

and spanwise curvature (refs . 42 t o 45) .

For simple isotropic panels, plate thickness, modulus of elasticity, and length are

usually com bined with f lutter dynam ic pressure into a single nond imensio nal

parameter .

h*f = 2 4 ( 1 - v 2 ) q f a 3 / E h 3

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where v is the Poissons rati o of th e panel material , a is the panel spa n, E is the m aterial

modulus of elastici ty, and h is panel thick ness. For such panels , f lu tter is characterized

by h *f exceeding some crit ical num ber determine d by aerody namic parameters . For

more complex panels, h*f is also a fun ctio n of th e remaining s tructura l param eters

(refs. 15 and 16 ).

The s implest procedure fo r mathematically modeling panels with elastic sup por ts will

normally be to judge the flexibil i ty of these supports by measuring the natural

frequencies (perhaps only the fundamental panel mode) and selecting the theoretical

support flexibility w hich will best m atc h the measured natural frequencies. Linear

structural theory is also used to determine the e ffects of in-plane mechanical or

thermal loads if they do not cause buckling. The degree of in-plane as well as

out-of-plane sup po rt condit ions is determined experimentally fo r such loads through

a

vibration test, the simp lest proc edu re, altho ugh a buckling test can also be used.

Nonlinear stru ctu ral theor y is required

to

predict th e natural frequencies of panels with

loads which cause buckling, panels with curvature in the direction of flow (which will

consequently have aerodynam ic preloading due t o their inh erent geom etry), or panels

under pressurization (refs. 1 5, 26 , and

46

to 48 ). This requirement is necessary because

there are sub stantial changes with changes in stress in the natura l frequencies of panels

subjected to a significant pre-flutter static stress. Structures sensitive to geometric

imperfections also require a nonlinear tr eatm ent (refs . 45 and 49 ). Nonlinear theo ry is

always required to predict th e l imit-cycle amplitude and s tresses of any panel th at has

penetrated into the flu tter regime.

Because no reliable theory is available fo r predicting str uct ura l dam ping , it can only be

determined f rom exper iment , e i ther by the decay or f requency-bandwidth method

(ref . 9) .

With regard to s t ruc tura l theory for or tho tropic panels (an im portan t considera tion for

many practical designs), the situation is som ewh at co mp lex. If a pane l is truly

orthotropic, then a well developed l inear s tructural theory

is

available f or d etermining

the panels natural modes and frequencies (refs . 26 and 32 to

40).

The corresponding

nonlinear theory, al tho ugh basically un dersto od (re f. 17), has not been applied t o the

panel fl utte r problem, and hence no capabil ity exists for handling buckling,

pressurization, or s treamwise curva ture of or tho trop ic p lates . An o r tho trop ic model of

a panel is usually acceptable if the wavelength of the fl utte r mod e (or distance between

nodal l ines) is large compared to the dis tance between st iffeners or other

discontinuities.

If an orth otro pic m odel is inappropriate , th e only recourse is t o use a more

complicated model which tre ats the s tru ctu re in terms of i ts individual comp onen ts .

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Fini teele me nt , f inite-dif ference, or comp onen t mode methods may be used for

analyzing these more comp licated models (refs. 8, 9, and 17). Th e principal cr iter ion of

success is the abi l ity t o co mp ute the natural panel modes and f requencies accurate ly.

For some s t i f fened panels the eccentr ic i ty

of

the s t if feners may be im por tant to i ts

f lut ter behavior . Although this parameter has been widely s tudied for i ts ef fect on

buckling, reference 50 is one of th e few wh ich discuss its effect on f lu tter .

I t is usually desirable t o verify th e theoretica l predictions of frequencies and mo des by

measurement . i f

the 'ilieoreticd mode

proves to be inaccurate, these measurements

may sometimes replace the theoretically predicted natural frequencies and modes in

the f lutter analysis. (See Section

2.4.

For some panels, the num ber of natural modes

required for an accurate f lutter analysis may be too large to measure in practice.

Ort hot rop ic panels or those with large length-to-width ratio are typical.

Finally, various types of panels can be ranked approximately in order of the precision

with which th eory an d/o r tes ts can predict the onset and severi ty

of

their panel f lut ter

oscillations. This is roughly the same ord er in w hich one can accurately determine the

panels ' natural mo des and frequencies. Th e main diff iculty lies in predicting panel

stiffness, and p erhap s the most diff icult parameters to evaluate are

1 )

variable stiffness

(e .g. , or thotropy or determinat ion of equivalent or thotropy

of

built-up panels) ; (2) the

effective stiffness of buckled plates;

3 )

curvature; and 4) anel boundary suppor t

conditions particular ly for variable-stiffness, loaded plates whose stiffness may be

sensi tive to su ppor t condi t ions .

An app rox ima te ranking of various panel types in orde r of their increasing diff iculty of

predict ion

of

panel flutter onset and severity is given in the following listing. In

constru cting this l ist we distinguish between geometric factors and types of panel

loadings.

Geom etr ic Factors

(a) F la t , i sotropic panels

(b) F la t , or thotrop ic panels

(c) Fla t, s tr inger-stiffened panels

(d) Isotropic panels wit h spanwise curvature

(e)

Isotropic panels with streamwise curvatu re


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Loadings

(a) In-plane loads below buckling

(b) Pressurization

(c) In-plane loads bey ond buckling

Nonlinear f lu t te r theory is required to de term ine the panel f lu t te r boundary for

isotropic panels w ith s treamwise curvature o r panels under pressurization or und er

compressive loads beyond the buckling load, and to determine such panels natural

modes and frequencies . Nonlinear flutter theory is also required for any panel

geometry or loading to calculate flutter s tress levels and frequencies . Nonrectangular

planform shapes will offer analytical difficulty with some theoretical m eth od s (e.g.,

Rayleigh-Ritz or G alerkin); however, the accuracy of t he basic th eory normally is no t

affected significantly.

2 2 2

Aerodynamic Parameters

There has been almost exclusive reliance o n theoretical me tho ds to evaluate the

aerodynamic parameters and assess their effect on panel flutter. At tem pt s have been

made t o evaluate aerodynam ic th eory by measuring aerody namic pressures over rigid,

sinusoidally def orm ed surfaces an d also over oscillating panels, fo r com parison w ith

theory (e.g., ref. 5

1 .

Thesc measurements , and the cons t ruc t ion of accurate models ,

have proven to be q uite d ifficult . T he exper ime nts nevertheless appear to have yielded

limited verification of the aerodynamic theo ry o r , a t leas t, they have no t inva lida ted i t .

Confidence in the theory is based largely upon airfoil experience and the indirect

evidence of flut ter results ; the aerodynam ic the ory ap pears basically sou nd .

There are essentially three levels of aerodynamic theory available: (1) a quasi-s teady,

two-dimens ional o r pis ton theory approp r ia te to h igh supersonic Mach num ber ( re fs .

29

and

30); (2)

an unsteady (linearized, inviscid) theory (refs.

48, 5 2 ,

and 53)

appropria te f rom zero up t o h igh supersonic Mach nu mbe r; and

(3)

an uns teady,

shear-flow theo ry which acc oun ts for the variable, mean-velocity profi le d ue to

boundary-layer effects (ref. 54). The last theory is generally most needed at transonic

t o

low

supersonic Mach num bers or when the re a re th ick bo undary layers . Th e f i rs t

theory is the sim plest bu t also has the sm allest range of app licability and henc e is th e

least accurate.

Th e second and th i rd theories offer sys temat ic improv emen ts and

include th e first or first an d second as special cases.

For very high Mach numbers or relatively blurit vehicle configurations,

one

must use

the aerodynamic variables (e .g. , Mach n um ber , etc .) appro priate to th e local flo w field

10


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over the pan el, which may be substantially different from the free-stream values. Th e

dynamic pressure, and density, the in-plane f low angularity, and for some f low

cond itions the b ound ary-layer velocity profile and thickness.

important aerodynamic parameters are generally the local values of Mach number,

The aerodynamic theory in any of its several fo rms is mo st reliable at high sup ersonic

Mach numb er and when the boundary layer is

so

thin that i t may be neglected. For

such f low conditions, the quasi-steady

,

two-dimensional aerodynamic theory is quite

accurate. A s the Xach iiiiriibcr

decreases to 2

or less, the quasi-steady o r piston-the ory

analysis no longer accurately pred icts the aerodyn amic forces on an oscillating plate for

the following reasons: (1 ) the three-dimensionality of the f low field becomes

important when (M2-1)lI2 times the panel aspect ratio is less than

1 ,

and (2) the

unstead iness of the flow field gives rise to significant phase shifts between aero dyn am ic

force

and panel defo rma tion , which can be accurately described only by a fully

unsteady theo ry. Such phase shif ts may give rise to negative aerodyn amic dam ping in a

given panel mode, which in turn leads to an instabili ty in that mode. This so-called

single-degree-of-freedom insta bility , wh ich usually on ly occ urs for

M

SP 8004 - Panel Flutter

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Transcript of SP 8004 - Panel Flutter