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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).
2
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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 .
4
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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
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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