THE UNIVERSITY OF MICHIGAN
COLLEGE OF ENGINEERING
Department of Aerospace Engineering
Final Report
PANEL FLUTTER OF CYLINDRICAL SHELLS
W. J. Anderson
ORA Project 08079
supported by:
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
GRANT NO. NGR-23-005-166
WASHINGTON, D.C.
administered through:
OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR
August 1967

PART A

AEROELASTIC STABILITY OF PLATES AND CYLINDERS
William J. Anderson
The University of Michigan
ABSTRACT
Linear stability criteria are presented for the panel flutter of
thin plates and thin-walled cylinders. These structures are exposed to
fluid flow passing parallel to an outer surface. The expression for
fluid pressure is simplified in order to emphasize the dynamic properties of the systems. The pressures are derived from steady flow relations (frequency effects are ignored). An arbitrary spatial phase angle
is included in the pressure expression. As this phase angle is varied
in a continuous manner, the fluid flow passes from "subsonic" character
to supersonic character. The results are useful in classifying several
types of instability and discussing several pathological cases which
are usually treated separately.
The analysis is intended to serve as an aid to understanding the
mechanism of panel flutter; however, it can be applied directly to
several problems. It is accurate for the static divergence and "coupled
mode" flutter of flat panels in supersonic flow, and also for divergence
problems wherever experimental measuremental meacan supply the values for
the necessary aerodynamic parameters. One result is to point out the
1

importance of static instability for flat panels in a transonic viscous
flow. A second result is to illustrate that the asymmetric divergence
of cylindrical shells is very sensitive to small changes in the pressure distribution.
2

A, A Aerodynamic pressure parameter; pUL pUR
D Eh3/12(l - v2) I
F Airy stress function
h Panel thickness
K Aerodynamic pressure constant, Eq. (5)
m
L Length of panel
M Mach number
m Axial wave number
N Number of modes
N Axial stress resultant due to initial load
x
NQ Circumferential stress resultant due to initial load
p(x,t) Aerodynamic load
q Integer
R Radius of cylinder
t Time
U Flow velocity
w Panel displacement in transverse direction
x Spatial coordinate, flow direction
z Spatial coordinate
&s ~ Kronecker delta
qm
Amplitude constant
Q Angular coordinate
X Eigenvalue
X Eigenvalue
5

p Fluid density
ps Panel density
xr Axial stress in cylindrical shell due to shell motion
|r* Spatial phase shift
w Frequency, rad/sec

LIST OF FIGURES
Fig. 1. Typical panel flutter problems.
Fig. 2. Flow over an infinitely long, stationary, two-dimensional
wavy wall.
Fig 3. Flow over a two-dimensional flat panel.
Fig. 4. Stability boundaries for a flat plate.
Fig 5. Stability boundaries for a cylinder.
5

1. INTRODUCTION
The elastic instability of thin panels exposed to fluid flow is
under intensive study at the present time. Typical problems involve
thin-walled structural elements with one surface exposed to fluid flow
essentially parallel to the surface. Figure 1 illustrates the flow
situation for a flat plate and a cylinder. The usual question of interest is whether the elastic panels incur divergence (static instability)
or flutter (dynamic instability) at some value of flow velocity.
The fluid pressures exerted on oscillating panels are difficult to
derive in many cases. The role of fluid viscosity, frequency of oscillation, and panel geometry have complicated panel flutter studies to the
point where the results are often difficult to understand.
The present study is based on an intuitive simplification of the
pressure distribution on the panel. It illustrates the effect of the
spatial distribution of pressures. The pressures are taken from steady
flow results and are hence independent of the frequency of oscillation.
The results are valid only for instabilities occurring at relatively
low frequencies.
An approximate solution is required because of the nature of the
assumptions on the pressures. These assumptions are equivalent to a
specification of the generalized forces on a discrete system. Galerkin's
method is used to pose the eigenvalue problem in matrix form.
6

2. FLUID PRESSURES
The pressure expression used in this study is motivated by the solution for flow over an infinitely long, two-dimensional stationary wavy
wall (Fig. 2). For the case of inviscid, isentropic flow, one finds
that a deflection
w(x) = E sin 2x (1)
yields a pressure of the form
p(x) = e 2Cos + (2)
where 4 takes the value 0 for a supersonic flow and t/2 for subsonic
flow. The solution is not valid near Mach 1.
The pressure expression given in Eq. (2) is "exact" within the
framework of linearized potential flow for the stationary wall under
consideration. We will view this expression, however, as an approximation which has been provided to describe a given physical situation:
a panel of finite length with viscous flow effects, real gas effects,
etc. As an example, for transonic flow, McClure[l] measured pressures
of the form
p(x) = EPU 2 K cos (2 ()
for a stationary wavy wall. The constants K and ~ are functions of
Mach number, fluid properties and wavelength. McClure found the amplitude constant K to be near unity. His measured values of.f ranged from
7

20 to 450. We hence see that values of r lying between 0 and 90~
have physical significance in practical cases.
Let us consider the pressure expression, Eq. (3) as sufficient for
our purposes. We will generalize this expression slightly by using subscripts to show the dependence of the constants K and r upon the wavelength. For a given deflection of a wall
N
w(x,t) = e am sin (4)
m = 1 n
one then has a pressure expression of the form
iot pU2 N m
p(x,t) = et m - am K cos (- r)
M L m m
(5)
Note that each term in Eq. (4) represents a wave with length 2.
In the following examples, it will be assumed that the constants
Km and *m are known. (This is equivalent to assuming that the generalized
forces are known for the discrete system.) For example, if slender
wing (Ackeret) theory were used for supersonic flow over a finite panel,
Eq. (5) would result with Km = 1 and 4m = 0 for all m.
3. FLAT PANEL OF FINITE LENGTH
Consider the case of a two-dimensional flat panel exposed to
fluid flow over one surface Fig. 3. The plate is of uniform thickness,
length L and simply supported at both ends. The aerodynamic expression
of Eq. (5) will be used to provide fluid pressures above the panel.
8

The fluid below the panel is at rest and at the same static pressure
as the upper flow.
The equation of motion for small deflections of the plate is
D N - w w + p+p(xt) = 0 (6)
6x4 Ox2 6t'
and the boundary conditions are
2 2
w(O,t) = w(L,t) = 2 o (0,t) =
ox2 - xt
The solution is assumed to be of the form
N
iut Nmrcx
w(x,t) = e am sin L
m= i
Galerkin' s method yields a set of coupled, linear algebraic equations
of motion
N 2
N 4 N xO2 2
Z" t) +.x. (mx) - mrAK sinrm - Sm (7)
m m D m
+ AKm cos mm am = 0 (q = 1,2,...N)
where
2 4
~ -- - 4m
m - q2 if m + q is odd
A = pU23 Tmq
l 2-p 1' D )0 if m + q is even
and tmq is the Kronecker delta.
This is a linear eigenvalue problem in the eigenvalue k. It is
non-Hermitian and hence in general we may have complex eigenvalues.
9

The characteristic polynomial is solved for the eigenvalue as a function
NxL2
of A and -x
D
To interpret the stability of the system, we must remember that
the frequency of oscillation varies as the square root of the eigenvalue:
1/2
CD oc /
and hence
1/2t
w(x,t) c eiX
The square root must be considered a multivalued function of the complex variable X. If all eigenvalues \ are real and positive, then neutral
stability results. If x is real and negative, static divergence occurs.
If x is complex, then flutter occurs.
Results have been calculated for the stability of a panel with no
membrane tension (Nx = 0). Extensive experience with Galerkin's method
as applied to fourth order differential equations has shown excellent
convergence when four modes are used. Two-mode, four-mode, and eightmode calculations were used here; the results were found to converge
adequately.
The stability boundaries shown in Fig. 4 are from a four-mode
analysis. For this special case, the amplitude constants and the spatial
phase shift have been set equal for all modes:
K1 = K2 = K3 = K4 = K
='2 = I3 - 4 ='
10

As a result, the amplitude constant is easily incorporated into the
ordinate. The figure hence emphasizes the role played by |.
The panel is' stable for sufficiently low values of A, regardless
of the value for 4. As A increases, however, the panel becomes unstable
at some critical value of A. This can be either divergence or flutter,
depending on the value of 4.
It is interesting that for 0 = 0 ("supersonic" flow) only flutter
is possible. (Experimental evidence indicates that this theoretical
solution is correct for r = 0.) Also, for * = 90~ ("subsonic" flow)
only divergence is possible. These limiting cases are well known. On
the other hand, for phase angles r between 25~ and 90~, one encounters
divergence first and then flutter.
The results for small values of X, say from 0~ to 40~ are important.
In transonic flow, for instance, * depends upon boundary layer thickness,
fluid viscosity, etc. If a given test were carried out for varying
boundary layer properties, the type of instability might well change
from a dynamic type to a static type because of this spatial phase shift.
(It must be remembered that the present analysis cannot predict the singledegree-of-freedom type of flutter which often typifies transonic flow.
On the other hand, this analysis is "exact" for simply supported plates
which diverge and hence is sufficient to predict static instability.)
For phase angles ~ near 90~, one finds that increasing dynamic
pressure causes first a static divergence, followed by dynamic instability
and finally a static divergence. This might be a confusing factor in
11

some subsonic experimental work, where spatial phase angles might be
near, but not exactly, 900.
4. ASYMMETRIC FJUTTER OF A CYLINDER OF FINITE LENGTH
The stability of a finite elastic cylinder Fig. 1 will be investigated in the same spirit as the flat panel. The shell is of uniform
thickness and unstiffened. Conventional cylindrical coordinates x,
r, 0 will be used. Donnell's cylinder equations are adequate to describe
the deflections of interest here:
2 2 + 1 2 + sh + p(x,t) = (8)
Rx2 R R x2 t2
V4 Eh a2 =
v - - 2 o (9)
R x2
The boundary conditions are taken to be the freely-supported case:
2
v = w = 2- = x (at x = 0, L)
ax
Again, for a deflection of the form
iat _ Wx
w(x,Q,t) = e cos nO sin x
L
the fluid forces will be taken as
If one again applies Galerkin' s method to the equations of motion
(8) and (9), one obtains a system of linear algebraic equations
1
p =9 4 IM _: elt (cos nO) Km -. cos mx )
12

m=l am L[L\2 + n 2 - v\h \ L ( L L J') [ 2 D \L
2 h
+ n2, AKM- R sin
NR n - -A - sin q m
PshaR24
2 - 1 D
and nqm is defined as for the plate.
These equations can be solved for the eigenvalues % as a function
of the fluid dynamic pressure ratio A and the phase shift r. We will
consider numerical results for a case corresponding to wind tunnel
tests carried out by Olson [2].
NX = 0
Nk =
R = 8.00 inch
h = 0.004 inch
Q = 15.4 inch
v = 0.55
n = 28
We will again choose
81 = 82 = *-'n = V
K1 = K2 = *. Kn = K
13

The results for a four mode solution are given in Fig. 5. Here
it is seen, as for a. flat plate, that for t = 0 only flutter can occur.
For values of 4 between 600 and 1200, there is an unexpected result.
The case of static divergence does indeed occur, but at relatively large
values of ~. In this case, if ~ is not exactly 90~, then flutter can
occur at much lower values of A.
This analysis shows the danger inherent in using an aerodynamic
theory which predicts that * = 90~ exactly. Resulting calculations
might not reveal a flutter situation which occur at a much lower dynamic
pressure ratio.
Note that the flutter boundary is very insensitive to changes in
t from -30~ to 60~. This means that the details of the pressure distribution on the cylinder are not of much importance in the stability analysis.
This explains why one of the simplest aerodynamic theories, Ackeret
theory, can be used with success to predict cylinder flutter which occurs
at low frequencies [3].
5 CONCLUSIONS
The appearance of a spatial phase shift as a free parameter in the
fluid pressure expression results in some new observations. It illustrates the change, in a continuous manner, from subsonic (or slender
body) flow character to supersonic character. The intermediate values
of the phase angle have physical application to the cases of viscous
transonic flow over flat plates and supersonic flow over cylindrical
shells.
14

The analysis is limited to two types of elastic instability:
coupled mode flutter and divergence. The study cannot predict single
degree-of-freedom flutter because of the use of steady flow relations
for the fluid forces.
Several examples were studied in which the pressure amplitudes Km
were identical in all modes and the phase angles rm were identical in
all modes. This case was chosen because of its simplicity. Conclusions
for the flat plate and the cylinder will be discussed separately.
The flat plate exhibits both divergence and flutter. For
one range of the spatial phase angle r (-90~ to -60~), the plate is
stable jor all dynamic pressures. For a second range of 4 (-60~ to 25~),
only flutter is possible. Finally, for a third range of t (25~ to 90~),
divergence is the critical form of instability, occurring at a much lower
dynamic pressure than flutter. The stability diagram indicates that
experiments carried out for certain phase angles might be confusing in
the sense that different regions of stability and instability could be
observed in turn as the dynamic pressure is raised.
Divergence occurs for flat plates at a relatively low value of
dynamic pressure ratio. As a result, divergence may be a distinct
problem for the case of viscous transonic flow, where previous pressure
measurements indicate that the necessary phase shift does occur [1].
The cylinder example studied was for a particular cylinder geometry,
chosen to match the only successful experiments to date. The cylinder
exhibits coupled mode flutter over the entire phase angle range of
physical interest. This flutter boundary is surprisingly insensitive
15

to the value of *. This is fortunate from a practical standpoint. It
means that coupled mode flutter calculations can be carried out for such
a shell with lessattention paid to the details of the spatial pressure
distribution.
The occurrence of divergence for the cylinder is not a simple phenomenon. In the past, divergence has been predicted for some types of
cylinders in supersonic flow (where axial wavelengths are long compared
to circumferential wavelengths). For the cylinder studied here the
divergence would be of little practical interest. Very small phase
shifts from 4 = 90~ cause flutter to occur at much lower dynamic pressures than divergence.
It is not prudent to extend the results of this simple analysis
too far. On the other hand, it can serve as a qualitative aid to investigators in panel flutter. There are times when the methods of analysis
are so cumbersome that one restricts his techniques (or his interest)
to only divergence or to flutter. It is apparent that one must be
careful to not overlook one of the possible instabilities.
6. ACKNOWLEDGEMENT
This investigation was supported by NASA Research Grant
NGR-23-005-166.
16

7. REFERENCES
1. McClure, J. D., "On Perturbed Boundary Layer Flows," Massachusetts
Institute of Technology Fluid Dynamics Research Laboratory Report
No. 62-2, Cambridge, Mass., June, 1962.
2. Olson, M.D. and Fung, Y.C., "Supersonic Flutter of Circular
Cylindrical Shells Subjected to Internal Pressure and Axial Compression," AIAA Journal, Vol. 4, No. 5, 1966, pp. 858-864.
3. Olson, M.D., "On Comparing Theory and Experience for the Supersonic
Flutter of Circular Cylindrical Shells," California Institute of
Technology Graduate Aeronautical Laboratories Report, AFOSR 66-0944,
Pasadena, Calif., June, 1966.
17

Fig. 1. Typical panel flutter problems.
18
~- <\ \' Ci

U
u
z
Fig. 2. Flow over an infinitely long, stationary,
two-dimensional wavy wall.
19

u
z
I////
T//L
Fig. 3. Flow over a two-dimensional flat panel.
20

0
oj
W....I
co to
^\ ~0 (/
ir~~~~~~
oJ
>c
Wi K) 4
w w
— 0 +
I-~~~~~~~~~~~(*L xiw (1)
Z Z ik
w,~'~
30N39~13^10 co
3 ~ C / 0^
I~"- --— ^ ^ ^ I I..J — S "^ 4 yC
C.
— 0 —- ^^^\ ~ l~y4s
0,:r
-^ w~..I _.. *
-5 I\ y h- ^
O0 q
1 <<
F —UJ
-J -I1
CO~~~C
Ir,./>lV OIJ.VHI 3~InSS3td 311WNYAN
21

250x 103200- w
2~ 5; 150 - >
E FLUTTER FLUTTER
50 TAB STABL
0 _1
Z ~,
>. 50
-90 -45 0 45 90 135 180 225 270
SPATIAL PHASE ANGLE ~, DEGREES
Fig. 5. Stability boundaries for a cylinder.

PART B

ENGINEERING ESTIMATES FOR SUPERSONIC FLUTTER OF
CURVED SHELL SEGMENTS
William J. Anderson and Kuo-Hsiung Hsu
The University of Michigan
ABSTRACT
Static aerodynamic theory is used to find design curves for the
flutter of curved panels. The panels are rectangular segments cut from
a circular cylindrical shell, Supersonic flow is directed parallel to
the generators of the shell segment. The pressure expression used is
general enough to encompass a wide range of physically possible pressure distributions. Design curves are given, in the form of a thickness
parameter required to prevent flutter as a. function of curvature and
length-to-width ratio, Upper and lower bounds for the onset of coupledmode panel flutter are given. Comparisons with other theories and experiments are madeo The results are intended to aid in design of wind tunnel
models for panel flutter tests.
25

D Eh2/[12(1-v2)]
F Airy stress. function
H Thickness para.meter, 7 V
h Panel thickness
L Length of panel
M Ma.ch number
m Axial wave number
N Number of modes
Nx,NQ Stress resultants, see equations (5) and (6)
p(x,,t) Aerodynamic load
q Integer, also dynamic pressure
R Radius
t Time
V Flow velocity
W Width of panel
eff Effective width of panel, W/n
w Panel displacement in radial direction
x Spatial coordinate, flow direction
LLi1 v2
Z Curvature parameter, h l-v_
6m Kronecker Delta
Q Angular coordinate
go QIncluded angle of shell segment
j< XEigenva.lue
p Fluid density
24

PS Panel density
\'V Spatial phase shift
co Frequency, ra.d/sec
25

LIST OF FIGURES
Fig. 1. Shell segment.
Fig. 2. Elastic shell segment imbedded in a cylinder.
Fig. 3. Thickness requirement for a cylindrical shell segment,
= 0~ (Ackeret theory),
Fig. 4. Thickness requirement for a cylindrical shell segment,
t 90~ ("slender body" theory),
Fig. 59 Upper and lower bounds for thickness requirement.
Freely supported edges.
Fig. 6. Comparison with other theories and experiments.
Fig. 7, Stability boundaries, - = 0~ (Ackeret theory), four mode solutiono
Fig. 8. Stability boundaries, * = 90~ ("slender body" theory).
Fig. 9. Approximate upper and lower stability boundaries.
26

1o INTRODUCTION
There is a. need for rough estimates of panel flutter boundaries in
design work. One specific area which has not been studied extensively
involves the flutter of a rectangular panel with curvature (Figo l)o
This panel would in general form a portion of a cylindrical body (Figo 2),
and would be supported at its edges by heavy bulkheads or stringerso
Some portions of the outer skin of a missile would correspond to this
case For conventional aircraft, such a panel might represent a window,
where the window is relatively weak compared to the surrounding structureo
Such windows can be a problem in high speed flight where temperature lowers
their rigidityo
The exact mathematical solution to this problem is so difficult,
and the results dependent on so many parameters, that there is serious
doubt whether it is of any practical value. On the other hand, recent
research [1] for cylindrical, shells indicates that approximate results
can be found by using a. steady flow (quasi-sta.tic) theoryo The approach
taken here is to recommend a. set of design curves developed by a simple
theory with the intent that corrections to these curves are to be made
a.s experimental data are obtainedo It is felt that the design parameters
used here are somewhat universal and will be the ones which will prove
useful in the long run, even after more precise theories are availableo
In the mathematical development of the problem, we will remain withy
in the framework of linear shell theory and steady flow theoryo A modal
27

approach will be used, hence the aerodynamic problem resolves to finding
pressures on sinusoida.lly deflected wallso The aerodynamic pressure
distribution used. in this problem will be generalized in the manner studied
in Refo [2]o" A parameter * is introduced to typify the spatial pressure
distribution. Then 4 is allowed to vary over the range of values which can
be expected for such a panel under different physical conditions including
boundary layer effects and length-to-width ratio effects. The result is
an approximate theory which gives upper and lower limits to the panel
thickness requirement to prevent fluttero The upper limit corresponds
to the use of Ackeret theory, the lower limit corresponds loosely to a.
"slender body" type of theoryo These two bounds represent extremely different flow situations, yet the dynamics of the system are so insensitive
to the details of the pressure distribution as to cause a variation from
upper to lower bound of only 355 for most caseso
A series of figures will be presented for design purposeso These
should be especially useful in designing models for wind tunnel testingo
Previous work has been done on related problemso Dzygadlo [3]
studied the elastic instability of an infinitely long ela.stic segment
of an infinitely long cylindero The stability boundaries were found for
a traveling wave form~
w(x,,t) - w(Q) eiK xVt) ). ~.i.-~. o i'
*Part A of this reporto
28

A set of integro-differential equations of motion resultedo These were
solved with the aid of a. Fourier series in the 0 variableo Much effort
was placed on a. study of the effect of structural (Voigt) damping on the
stability boundaries. For moderate amounts of damping, unexpected changes
in the panel's stability resultedo The numerical results presented were
not extensive It was concluded that for small damping ratios and for
fixed shell thickness and radius, the critical Ma.ch number does not vary
greatly for included angles for the segment lying between /t/4 and jto
Another study of interest was by Dowell and Widnell [4]. The case
considered was a finite length elastic segment in an infinitely long
rigid cylindrical shello In this case, the generalized aerodynamic forces
were found for deflections of the type
- gid Rigid i
Rigid i wt v /
( @ t) i0it n i mx
w(x,0,t) = e cos nO sin -LDowell made several comments about the stability of the shell segment
merely by looking at the character of the generalized forces~ First of
all, in the low supersonic Mach number range, a. single degree of freedom
type of flutter is possibleo Secondly, for shell segments with long
length-to-width ratios, static divergence takes placeo Flutter boundaries
for the "coupled-mode" type of flutter were not presentedo
Neither of these studies is easy to extend to the current problem.
Neither case yields useful design curves (nor were they intended to)o
The approach used by Dowell would be the more easy to extend to the present
ca.se~
29

The present solution parallels the approach used by McElman [5] to
some extento McElman studied a curved orthotropic panel segment by using
a two mode analysis with Ackeret theoryo No design curves of the type
shown here were presented in McElman's workO (In order to work with lower
aspect ratio panels, one needs many modes rather than twoo)
2. STATEMENT OF PROBLEM
Consider a cylindrical shell segment as shown in Fig. lo Supersonic flow passes over the outer surface of the segment, with flow direction parallel to the cylinder axiso The segment is of uniform thickness
and of isotropic, homogeneous elastic material~ Conventional cylindrical
coordinates x, r, Q are usedo The shell segment is defined by
r R
0 < x < L
go0 < ~ < Qo
2 - ~ 2
Deflection of the surface of the segment will. be given by w(x,,t) measured
from the mean radius of the shello The edges of the shell will be "freelysupported" as defined belowo The shell may be internally pressurizedo
No structural damping will be includedo
STRUCTURAL DETAILS
The shell is thin and initially circularo Radial deflections are
restricted to be small~
50

W(x,.t) << 1.
h
The in-plane motions of the shell u(x,Q,t) and v(x,Q,t) are small compared
to w(xQt) so that inertial effects due to in-plane motion can be neglected
(Reissner's assumption) The included angle go will be less than Tt/2 so
that Donnell's shallow shell equations can be used,
Dv4w 2x Rw N0 a2W 1 w2F a2w
DV47 + - + ph + P(x,^,t) = 0 (1)
V4~ Eh a 0 (2)
R ax2
where D is the bending rigidity of the shell, Nx and NQ are constants
representing the components of membrane stress due to internal pressurization and F(xQt) is the stress function defined so that
Nx(xet) = R2 a (3)
(4)
NW(x,Qt) 2 (4)
6x2
Note that 5v and NQ are the time dependent components of membrane stress
due only to panel motion,, The total membrane stresses are
Nx(x,~,t) = Nx + Nx(x,9,t) (5)
N (x,Qt) = Q + N~(x,Qt) (6)
Boundary conditions to be applied at x = 0O x- L are
V'= W = ~ = = 0
1 F (7)
v - w OA

Boundary conditions at Q = ~ are
2 2
w - u = = ~ (8)
a ax
These freely supported boundary conditions have been chosen primarily
because they are satisfied (term by term) by the series
iwt N n:, m x 0 < x < L
w(x,Qt) = e, am cos -o sin - o ~ — \
m=lm go -G W
\2 < < 6 (9)
These boundary conditions are useful, however, because they result in a
dynamically "weak" plate, Hence, the stability boundaries will tend to
be conservative for design purposes when applied to damped plateso
At this point, the structural problem has been posedo We need to
find the aerodynamic pressures p(xOt) generated at the panel surface.
AERODYNAMIC DETAILS
A strong assumption on the aerodynamic pressures will be madeo The
pressure on a panel deflection
w(xOt) eiot cos nQo sin m-x (10)
Qo LL
will be assumed to be
p(xQ,t) = C e p2 m- cos n cos + ( (11)
~^T2 L Lo 1
In other words, the pressure will have a magnitude equal to that given
by Ackeret theory and a spatial pressure distribution that can vary as
desired (It would be possible to discuss this same assumption later
in terms of generalized forces, but this is not as meaningful.)
32

The above assumption appears to be a strong oversimplification at
first glance. One wonders how to choose a proper value of * for a cylindrical shell segment. In Reference 6, extensive numerical work was
done for pressures on oscillating cylinders exposed to potential flow.
The phase angle r was found to vary only between 0~ and 90~. Hence, we
will include values of \ between 0~ and 900 in the present study. In
Reference 2, the dynamic results of such an assumption are studied in
detail. It is found that the choice of' does not drastically affect
the thickness requirement for cylindrical. shells.*
STABILITY DETAILS
Ga.lerkin's method is used to pose the problem in matrix form. The
deflections of the shell segment are
w(x,Qt) eiwt nros C a sin x (12)
w ~) =e cos - m m sin mx1
Note that this expression allows n half waves in the circumferential
direction of the panel. If n takes a value higher than 1, then the effective length-to-width ratio of the panel increases accordingly because
there are stationary nodal lines down the length of the panel.
*Footnote: The thickness required to prevent flutter is a continuous
function of $. For the case studied in Ref. [2], the thickness requirement has a minimum near V = 30~. This value of the thickness ratio at' = 30~ is practically identical with that at' = 0~. Because the
calculation for' = 0~ has more physical meaning (Ackeret theory) it
is used as a. reference rather than' = 30~.
335

The expression for pressure, Eq. (11), is used in conjunction with
Eqs. (1) and (2) to yield the set of linear algebraic equations of motion:
El^ a, _+ 2m2_ _ + _ L _
Z ~m,[m + (_L)2]2 1'' n NlXL 2 N.......
j2 mQ mWeff i4m2+L (L ~ t2D?t-D Weff
Weff
A 24 sin i +24 cos = 0
s4H3- qm, 4'm jW qm
(q = l,2,..n)
whereo
H = M2-M E h
(l-v2) q L
R h
L/Weff Ln
4A
^ PS ohw L
D
0 m+q even
qm 1-qm m+q odd
-m
Thus, a set of linear algebraic equations are obtainedo The occurrence
A
of a negative eigenvalue k signifies static divergence of the panel and
A
complex A signifies fluttero
35 RESULTS
Stability boundaries have been calculated for the aerodynamic loading
discussed aboveo All results will be given for cases with zero membrane
stresses Nx and Ig o This theory would be more inaccurate at positive
34

values of membrane stress which would cause higher flutter frequencies.
The results a.re presented using the effective length-to-width ratio
L/W eff a curvature parameter Z and a thickness parameter Ho The plots
of H versus L/Weff are given as a generalization of the work of Kordes,
Tuovila, and Guy [7], and the curvature parameter Z is chosen to correspond to Batdorf's study of cylinder buckling [81.
A four mode solution for = 0(Ackeret theory) is given in Fig. 35
It is easily seen that curvature helps to stiffen the panel and reduce
the thickness requirement. An interesting effect is obtained in the
regions where H increases with increasing L/Weff. This means that a panel
of given physical length and width will flutter in a mode with n > 1, giving
a higher critical value of L/Weffo As an example, a panel of length 10
inches and width 2 inches has a physical length-to-width ratio of 5. If
Z = 8000 for this panel then it must have a thickness ratio of H = Oo065
to prevent flutte:r from occurring at an effective length-to-width ratio
of 15. This particular panel flutters with n = 3, ioe., it has two interior nodal lines extending down its length~
Results for r = 90~ are given in Figo 4o These results are somewhat similar to the r = 0 curves except that the instabilities in the
lower left corner are due to static divergence. Again, one must observe
the cases where H increases with L/Weff and one must choose the multiple
of the geometric length-to-width ratio which gives the critical value
of Ho
35

Finally, several of the curves for 0 = 0~ and t = 90~ are combined
in Fig. 5. These are approximate bounds for the thickness required to
prevent an instability. Note that the difference between the two bounding curves is not great, particularly in certain intermediate regions of
L/Weff This may be an indication as to why Ackeret theory gives relatively
good results for the cylinder experiments discussed in Refo 1o In these
tests, a cylinder fluttered in a mode which yields an effective L/Wff =
9,21 and with Z = 6,950. Flutter occurred at H = 0,0624.
It is felt that figures such as Fig. 5 can be very useful to designerso
The curves give rough bounds for the onset of flutter or divergence for
freely supported panels. As experiments are carried out, confidence can
be obtained for accuracy of such curves.
It is suspected that for L/Weff large, more modes are needed to
ensure convergence. Gaspars and Redd [9] studied carefully the number
of modes required for convergence on finite aspect ratio flat plates
when Ackeret theory is usedo They found that as many as 50 modes were
needed for flat plates with aspect ratios of 10 or more. The present
results a.re less sensitive to convergence problems because of the presence
of curvature and because the flutter parameter H is less sensitive to
error in the eigenvalue of the matrix problem.
Other theories and experiments are shown in Fig. 6. Several of the
points shown correspond to work for full cylindrical shellso The problem of a shell segment is closely related to that of a full cylinder,
Structurally, the major difference is that the full cylinder can flut36

ter in modes with waves travelling in the circumferential direction whereas
the segment cannot. Of particular interest in Figo 6 are the experimental
points found for -full cylinders by Olson and Fung [1] and by Stearman,
Lock and Fung [10]. It is now suspected that these cylinders did flutter
in circumferentially travelling waves [11]. This might explain why the
experimental values occurred at slightly higher values of thickness ratio
than predicted by the present theory. The experiments of Tuovila and
Hess [12] were carried out for a shell segment clamped all around. The
tests were done at Ma.ch.1,3, which unfortunately brings in transonic effects into the comparison. In transonic flow the unsteady aerodynamic
terms a.re of importance and there effects are neglected in this theory.
The theories of Voss [15] and Shulma.n [14] both were done for a
complete cylinder with the use of Ackeret theoryo These should (and do)
correspond with the present calculations and serve as a. check.
The theory of Dyzgadlo [15] was carried out for a more exact aerodynamic theory on a finite length cylindero These were mode calculationso'hese appear to yield values of H slightly higher than the current work
which may reflect the fact that fewer modes were used by Dzygadlo.
(Gaspars and Redd [8] indicate that the thickness requirement decreases
with an increase in the number of modes.)
All in all, there are no unclassified experiments known to the
authors which furnish the proper comparison with the theory. Such tests
would be usefulo
Figures 7-9 are cross plots of the same data given in Figs. 35-5
37

4. CONCLUSIONS
Design curves have been given for prevention of aeroela.stic instability of curved shell segments. The calculations are approximate in the
sense that unsteady aerodynamic effects are ignored and because a. modal
approach was usedo These are the very reasons that the results a.re understandable, however~ From a practical standpoint, these design curves,
as corrected by experiment, will probably be more useful than exact theories.
The only case illustrated here was the case of freely supported edgeso
The results should be conservative if applied to panels with calmped edges.
One shortcoming of the current calculations is the limited number
of modes usedo Only four mode solutions were carried outo If more modes
were used, the results presented at higher values of L/W ff would become
more accurate.
38

5. REFERENCES
1. Olson, M. D. and Fung, Y. C., "On Comparing Theory and Experiment for
the Supersonic Flutter of Circular Cylindrical Shells," AIAA 5th
Aerospace Sciences Meeting, AIAA Paper No. 67-77, New York, January
1967.
2, Anderson, William J., "Aeroelastic Stability of Plates and Cylinders."
The University of Michigan Report 08079-2-F, Part A.
5. Dzygadlo, Z, "The Problem of Aeroelasticity of a Cylindrical Panel
and a Plate Strip Taking into Consideration the Transversal Coupling,"
Proceedings of Vibration Problems, Warsaw, 2, 5 (1964).
4. Dowell, E. H. and Widnall, S. E., "Generalized Aerodynamic Forces on
and Oscillating Cylindrical Shell," Quarterly of Appl. Mech., XXIV,
1, April 1966,
5. McElman, J. A., "Flutter of Curved and Flat Sandwich Panels Subjected
to Supersonic Flow," NASA TN D-2192, Langley Research Center, April
1964.
6. Anderson, William J., "Oscillatory Pressures in a Simplified Boundary
Layer with Application to Cylinder Flutter," AIAA Journal, Vol. 4,
No. 5, May 1966.
7. Kordes, E. E., Tuovila, W. J., and Guy, L. D., "Flutter Research on
Skin Panels, NASA TN D-451, Langley Research Center, Sept. 1960.
8. Batdorf, S. B., "A Simplified Method of Elastic-Stability Analysis for
Thin Cylindrical Shells, " NACA Report 874, Langley Research Center,
1947.
9. Gaspars, P. A. and Redd, B., "A Theoretical Analysis of the Flutter
of Orthotropic Panels," NASA TN D3551, Ames Research Center, August
1966.
10. Stearman, R. 0., Lock, M. H,, and Fung, Y, C., "Ames Tests on the
Flutter of Cylindrical Shells," GALCIT Structural Dynamics Report
SM 62-37, December 1962.
11. Private communication with M. Olson and D. Evensen.
39

12. Tuovila, W. J. and Hess, R. W. "Experimental Investigation of Flutter
of Buckled, Curved Panels", NASA Memo 5-18-59L, Langley Research
Center, June 1959.
13. Voss, H. M., "The Effect of an External Supersonic Flow on the Vibration Characteristics of Thin Cylindrical Shells," Journal of Aerospace Sciences, Vol. 28, No. 12, December 1961.
14. Shulman9 Y., "Vibration and Flutter of Cylindrical and Conical Shells,"
OSR Technical Report No. 59-776 Massachusetts Institute of Technologyy
June 1959.
15. Dzygadlo, Z., "Self Excited Vibration of a Cylindrical Shell of Finite
Length in Supersonic Flow," Proceedings of Vibration Problems, Warsaw, Vol. 35 No. 1, 1962.
40

WyRe0
Fig. 1. Shell segment.
Fig. 2. Elastic shell segment imbedded in a cylinder.
41

o o
o 0
//l
CD
If Hn
r-,= 0 ~. 0 -
o o0 o0 o
0 0\0 0\ 0 Cc
0 V\ \\-0 0 t
0^^ 0
_.0 I-I O I
o0,.e0
w t2 5U o~
II I I I I o
42
0 tD WU a) N 0
d 0
rhI-c2

THICKNESS RATIO, H L I, i
o0 O
o o ooo 2 2
o0 0 I I0I''
0 0
0 0
0
C ^ \\\ \
=0~i~DO\ \
&i \r \ ^mu \ 0
m m:\- \, N
1' W I
o < C'~ /
ti-< /iA <
(D g.T m -,', -'. I /
* 90 -I /
C~r'r
H /
IQ II
_ II r, II
CD o 0
(: I I
R. - I
r * \ V,'-1'
m 00:z /1 I I. ^: 5 0 —I I 00'
_J_____________-J "

/ / V-0
tCj
_0 / / / 2z
W OD
0 0 / /
OO:: Q)
o II I.'\ 1'I i -
z *
00303 o N0,,'0) 0*,,,, U0
S=o a / / 1)
) / II -- 1 N I )
~b/ / / N0, — (r rn
Q-/ /F
/I0>- ** I 0
Io -- \ 4
aP: 0W 0
0 V))_ H
/ - g \UNI I P
//I o~
~~~-0~~~~~~ ~ d
V)^ \ \ -u" U. 1
NHj ~Hr
-0J
CC c\ 0
6 o o0
0, - b 3/-A = H'OIiVd SSBN>43IH1
44

0
1i j
0I I
0 0I -
I0l I I I —
-, (d
II I
~~~~~~00 1'~ ~~ ~ ~~ 10
- o
I0 ~ I I c\ i'' / / \ I \ -0
0/ /\ _0 \ g
\/\ \
I U \
/O (\O L O
-/I b( -I)
Nqq q
IHOI.LVU SS3N131H1.
+[ W HJ;`]=
44j~~~

0.15
c!
l 1
I I
0
Weff,20
w 0.05
0.0
0 10 20 30 40 50x1
CURVATURE RATIO, Z = - R /
R h
Fig. 7. Stability boundaries, 4t = 0~ (Ackeret theory), four mode solution.
46

. — Flutter
-—.- Divergence
-U 0.10
I?0 I
I~^ I
0,.) We\ I - IL/W; I
0 10 20 30 40 50X103
eff
CURVATURE RATIO,
0 10 20 30 40 50x103
g. 8 Stability boundaries = 9 (slender body" theory).
47

. —--- _' 0 ( Flutter)
-- --- go90~ ( Flutter)
-- -.. aJ=90~ (Fl utter)
0.15
CIiI
0 cr
<n, I\ ^ L/Weff4 20
LCU 0.05RA, z
/Wef'4' -
L- -off =4 - -- -.
0 10 20 30 40 50X1O
CURVATURE RATIO, Z = -
Fig. 9. Approximate upper and lower stability boundaries.
48

HIII IIII II IUNIVERSITY OF MICHIGAN
3 9015 02499 5659