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FLUTTER OF SANDWICH PANELS AT
SUPERSONIC SPEEDS
By
Melvin S. Anderson
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute
in candidacy for the degree of
DOCTOR OF PHILOSOPHY
in
ENGINEERING MECHANICS
June 1965
FLUTTER OF SANDWICH PANELS AT
SUPERSONIC SPEEDS
by
Melvin S. Anderson
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute
in candidacy for the degree of
DOCTOR OF PHILOSOPHY
in
ENGINEERING MECHANICS
June 1965
Blacksburg, Virginia
- 2 -
II. TABLE OF CONTENTS
CHAPTER PAGE
I. TITLE . . . . . .
II. TABLE OF CONTENTS .
III. LIST OF FIGURES .
DT. INTRODUCTION. .
V. SYMBOLS .
VI. ANALYSIS. Statement of Problem a.nd Assumptions. Development of Governing Equations ...
VII. RESULTS AND DISCUSSION ........ . Simply Supported Leading and Trailing Edges . Clamped Leading and Trailing Edges .. Comparison with Two-Mode Solution .
VIII. CONCLUDING REMARKS ..
IX. SUMMARY • • • •
X. ACKNOWLEDGMENTS
XI. BIBLIOGRAPHY. •
XII. VITA ....•. . . . . .
1
2
3
4
6
9 9
11
37 37 39 45
49
51
52
53
55
- 3 -
III. LIST OF FIGURES
FIGURE PAGE
1. Sandwich Panel and Internal. Forces . . • . . . • • . • . . 10 a. Coordinate System. • . . • . • . . . . . . . • . . . 10 b. Forces and Moments Acting on Differential Element. . 10 c. Contribution of Nx to equilibrium in z
Direction. • • • • . . • . . . . . • • . . . • 10
2. Schematic Dia.gram of Frequency Loop. Dynamic Pressure Para.meter A Versus Frequency Parameter ~· • . . • 25
3. Flutter Values of Dynamic Pressure. Simply Supported Pan.el, kx = 0. . . . . . . . . . . . • . . . . . . . . . 38
4. Frequency Loops for Simply Supported Panels. 4o
5. Mode Shape at Flutter for an Unstressed Square Sandwich Panel . a/b = 1 kx = O. . . • . • • . . . . • . . . . . 41
6. Effect of Shear Stiffness on Flutter of Sandwich Panels. . 42
7, Effect of Shear Stiffness on Frequency Loops, Clamped and Simply Supported Panels. a/b = 1 kx = 0. . • . . . . . 43
8. Variation of Flutter Parameter with a/b Simply Supported Panels ..•..•..
for Clamped and 46
9. Comparison of Flutter Boundaries from Two-Mode Gal.erkin Solution and Exact Solution. kx = 0 • • . • . • . . . . 47
- 4 -
IV. INTRODUCTION
Since airplanes and rocket vehicles first exceeded the speed of
sound, panel flutter has become an increasingly important problem.
Panel flutter is the unstable oscillation of the external surface of
a flight vehicle caused by certain aerodynamic loadings in combination
with the elastic and inertial forces acting on the panel. The ampli-
tude is limited by nonlinearities although the panel may be destroyed
before this amplitude is reached.
Because it is such a recent problem, most panel flutter investi-
gations have taken place during the last ten years. The basic theo-
retical treatment of panel flutter was done in this country by
Hedgepeth in 1957 (ref. 1). A similar analysis by Movchan (ref. 2)
appeared in the Russian literature about the same time period but was
not generally known in this country until a couple of years later.
Both of these analyses used simplified aerodynamic theory that resulted
in closed form solutions to the problem. Hedgepeth used the two-
dimensional. static aerodynamics given by Ackeret while Movchan used
piston theory. However, both aerodynamic theories are of the same
mathematical form and give essentially the same result except at
Ma.ch numbers near 1. For some time it was believed that such
analyses were applicable for only a small range of Mach number and
panel proportions and therefore more accurate analyses appeared such
as that of Cunningham (ref. 3) who used three-dimensional. unsteady
aerodynamics. Because of the complexity of such analyses, numerical
- 5 -
methods must be used and there is difficulty in getting convergence
for certain parameters. Recently Bohon and Dixon have shown in
reference 4 that the use of the more exact aerodynamics is not
necessary in most cases, and that satisfactory flutter results can
be obtained for all panel proportions using the simplified aerodynamics
except at Mach numbers less than 1.3 where even the more exact theory
is in doubt.
Up to now practically all panel flutter analyses made use of
classical thin plate theory which does not account for shear deflec-
tions. This assumption is good for solid plates but can lead to
error for built-up panels such as sandwich plates. In the present
paper a flutter analysis will be made for an isotropic sandwich panel
in order to show the degradation of the flutter boundary as a func-
tion of shear flexibility. A two-mode solution to this problem has
been made by McElman (ref. 5). However, it is known that a two-mode
solution is not very accurate for a plate rigid in shear, hence this
solution could be expected to give only qualitative trends. Accord-
ingly, an exact closed form solution to the problem is obtained
herein within the framework of the two dimensional static aerodyna.mic
approximation. Because of the demonstrated accuracy of this
aerodyna.mic theory, numerical results obtained from the solution are
expected to be quantitively correct.
A,B,C
a
B=
b
c =
D
d =
M
m
n
- 6 -
V. SYMBOLS
constants used in solution for w, Qx, and G.y' respectively
92 [ 2 + kx (1 + r) + r ~
1 - risc panel length
1 - rkx c i + qJ c1 + rD
panel width
bending stiffness of isotropic sandwich
constants defined after equation (30)
t -1 5 an -T)
stress coefficient for axial load
Mach number
bending moments in sandwich panel
twisting moment in sandwich panel
n2D
roots of determinantal equation resulting from equation (16)
inplane normal loadings per unit length positive in tension
inplane shear loading per unit length
integer
p
q
r,r'
s,T,u,v w
x,y,z
a.,a,e,
13 =
r =
- 7 -
lateral loading on panel
transverse shear forces per unit length
dynamic pressure of air stream rc2D rc2D
shear stiffness para.meters r = b2DQ, r' = a2IQ
constants defined after equation (38)
lateral deflection of panel
panel coordinates
defined in equation (25)
~ ~ rc9
(1 + r)92 r
r panel density per unit area
ri= s+a. 9 =
A=
v
s =
p =
T
a/b 2qb3
dynamic pressure parameter 'A = -, 'A' 13D
2qa3 =-
13D
critical or flutter value of dyna.mic pressure parameter
/..(1 + r) 93 1 - rkx
Poisson's ratio
s - a. time
- 8 -
q:> =
<l>n = value of q:> at nth natural frequency
Notation
rotation of line element originally perpendicular to middle surface
1 + r - r~i 2
circular frequency
commas denote differentiation with respect to the following subscripts
-9-
VI • ANALYSIS
Statement of Problem and Assumptions
A rectangular isotropic sandwich panel is considered exposed on
one surface to a supersonic stream with the flow direction parallel
to one pair of edges. (See fig. l(a).) The strea.mwise edges are
simply supported while the leading and trailing edges are either both
simply supported or both clamped. The panel may be loaded by uniform
inplane direct forces but the inplane shearing forces are zero. The
problem is to determine for a given panel what flow conditions will
cause divergent motion or flutter. The following assumptions are
made in solving this problem: (1) The panel behavior is described
by the sandwich theory of Libove and Batdorf given in reference 6.
The essential feature of this theory is that straight line elements
perpendicular to the undeformed middle surface remain straight and of
the same length but are not necessarily perpendicular to the deformed
middle surface. This change in angle from a. right angle which is the
transverse shearing strain is given by the shear force divided by the
shear stiffness DQ. (2) The aerodynamic loading is given by the
two-dimensional static approximation (Ackeret theory, see ref. 7).
Thus the total lateral load on the panel, including inertia effects,
is:
2q p = - ~ w,X - 7W,TT (1)
where q is the dynamic pressure of the stream, ~ =JM.2-1 where M is the
- 10 -
z
y
x
(a) Coordinate system.
( Nx + Nx, x) dy (Ny+ Ny,yldx
(b) Forces and moments acting on differentiaJ. element.
(c) Contribution of Nx to equilibrium in z direction.
Figure 1.- Sandwich panel and internal forces.
- 11-
Mach number, r is the mass per unit area of the panel, T is time,
and w is the lateral deflection. The aerodynamic loading term in
equation (1) was derived in reference 7 by neglecting any entropy
changes, and by assuming the flow follows the surface and is turned
only through small angles. With the above assumptions, the analysis
proceeds as follows.
Development of Governing Equations
Differential equations.- The equations governing the motion of an
isotropic sandwich panel can be obtained by consideration of the
equilibrium of the forces and moments acting on an element as shown
in figure l(b). To avoid conf'usion, only the forces and moments
which act on the front sides of the element are shown. The forces
and moments which act on the opposite sides differ by only infinitesi-
mal amounts from those shown but do act in the opposite sense to the
corresponding force or moment on the front side. For example on the
rear face x equals a constant, a force Nxdy is acting in the
negative x-direction. Summing forces in the x- and y-directions
given the following two force equilibrium equations:
N + N = 0 x,x xy,y (2)
N + N = 0 xy,x y,y (3)
In summing forces in the z-direction it is necessary to determine the
contribution of the midplane forces as is illustrated in figure l(c)
- 12 -
for the Nx force. Similar considerations for the other two membrane
forces results in the following equilibrium equation for forces in
the z-direction.
~ x + Q __ y + (Nx w x) x + (Ny w y) y + (Nxy w x) y ' 7, ' ' ' ' ' '
(4)
By virtue of equations (2) and (3) this may be written as
(5)
Equilibrium of moments about the y and x axes requires that
M - M___ - Q = 0 ·-x,x --xy ,y x ( 6)
-M +M -Q =0 xy,x y,y y (7)
A relationship for the moments in tenns of the deflection w and the
two shear forces can be obtained from reference 6 in the following
form for an isotropic sandwich
(8)
M = - D [w - _L G. + v (w - J_ ~ _\ l Y ,yy DQ -y,y ,xx ~ ,x)J ( 9)
( 1 - v) D [ 1 ( 1 " Mxy = 2 2 w,xy - DQ \Qy,x + Qx,yLJ (10)
- 13 -
As DQ approaches infinity the terms containing the shear
forces approach zero and in the limit equations (8) through (10)
correspond to the familiar moment curvature relationships for plates
rigid in shear. Substitution of the expression for lateral loading
given in equation (1) into equation (5) and substitution of equations
(8) through (10) into equations (6) and (7) results in the following
three differential. equations in the three unknowns, w, 'tx and ~·
Qx - w,xyy - w,:xxx - -D- - l + v J ~,yy+ ~,xx+ 2 ~,xy = 0
(11)
- w - w - ~ - J_[l ,xxy ,yyy D DQ
+vQ +Q +1-vQ J=O 2 x,xy y,yy 2 y,xx
The inplane shearing force has been assumed zero so does not appear
in equation (11).
Bounds.ry Conditions.- The boundary conditions appropriate for
this set of differential. equations a.re discussed thoroughly in
reference 6. The boundary conditions are such that a boundary force
(moment) or the associated displacement (rotation) must be specified.
Thus at an edge x equal ccnsta.nt the boundary conditions may be
written
where
- 14 -
or w = O
or cp = 0 x
or Cf>y = 0
Boundary conditions at an edge y equal constant can be
(12)
obtained by interchanging x and y in equations (12). The basic
difference between boundary conditions from classical plate theory
and sandwich plate theory is that three boundary conditions must
be specified on each edge of a sandwich plate whereas only two are
required in classical plate theory. The increased number of boundary
conditions is related to the fact that the system of differential
equations for sandwich plates (eq. ll) is sixth order while the
equations of classical plate theory are fourth order. The extra
boundary conditions allow the load or deformation at the boundary to
be specified in greater detail. For example, it is not necessary to
use a Kirchoff shear boundary condition for a free edge of a' sandwich
plate since the transverse force, bending moment, and twisting moment
can be individually set equal to zero.
- 15 -
For the problem considered herein, the edges at y = 0 and
y = b are simply supported which immediately leads to the two
boundary conditions familiar in classical plate theory, namely:
w(x,o,T) = w(x,b,T) = 0
(13)
For the third boundary condition either the twisting moment ~ or
the rotation ~x must be zero. In order to have Mxy zero on an
edge, a sandwich panel would have to be supported only along its
midplane which is not likely to occur in practice. Thus the more
realistic boundary condition is that the rotation ~x be zero.
Inasmuch as w is zero along this boundary the third boundary
condition may be written as
Q (x,o,T) = Q (x,b,T) = 0 x x (14)
For the boundary conditions given by equations (13) and (14)
a closed :form solution to the differential equations can be obtained
as is illustrated in the next section.
General solution of differential equations for simply su:pported
streamwise ed.ges.- Expressions which satisfy the differential
equations (eq. (ll)) and the boundary conditions (eq. (13) and (14))
are:
- 16 -
1fY m.x iIDT w = Re A sin - e""!r' e b
~ = Re B sin ~ ellf eiIDT
~ = Re C cos ';;" emt eiwr
(15)
where m is the circular f'requency of the panel motion. The argument
in the trigonometric f'unctions in equation (15) could have been ta.ken
more generally as ll.ICL. However examination of such a solution reveals b
that it is equivalent to the present solution for a panel with a b width -. For most practical cases n = 1 results in the most n
critical flutter condition but results for any other value of n can b be obtained from the present anaJ.ysis by replacing b with n'
Substitution of equation (15) into equation (11) and simplif'ying
results in
A[s2 - m2] + Ba.b2 r ~ + ca2b [- 1 + 1 - v r m2 rl = o (16) rc2 rc3n re rc3n 2 92 rc2 - j
A - e - - + -- - s - r8 + r - - r - = o ~ 2 m3J Ba.b2 [ 2 1 - v 2 m2J Ca2b 1 + v m re rc3 rc3D 2 rc2 rc3D 2 re
- 17 -
a. a = -b
In order for a nontrivial solution to equation (16) to exist, the
determinant of the coefficients of A, B, and C must be zero.
Setting this determinant equal to zero gives an expression for
(16)
determining m. Actually six values of m are found this way since
as indicated in reference 6 this problem can be reduced to a single
sixth-order dif:ferential equation. Four o:f the values of' m are
found from the :following algebraic equation
where
m4 - 4SJn3 + ~ Am.2 + ~ - 1C4 B = 0
E =------4~ (1 - rkx)
A = 92 C2 + kx (1 + r) + rq>J l - rkx [
~ = A(1 + r) a3 1 - rkx
(17)
- 18 -
The roots of equation (17) will be labeled m1 through ml+· The
remaining two roots are given by
~,6 = ± • 9 / 1 + -( 1-~-v-)-r (18)
Since equation (16) are homogeneous, unique solutions do not
exist for the constants A, B, and c. However, any two can be
expressed in terms of the third. For this problem A and C will
be determined as a function of B. Corresponding to each mi there
will be a set of constants At, Bi, and Ci. They are related by the
following
r (~2 e2 _ m2) + ~ e2 i ab2 B i
i = 1, 2, 3, 4
i = 5, 6
Thus the general solutions for w, ~' and ~ are
(19a)
(19b)
w = Re ei(J.)T
- 19-
6 Qx = Re e iwr sin : I
i=l
(20)
Simply sup;ported leading and trailing ed.ges.- For simple supports
at x = 0 and x =a the boundary conditions (see eq. (12)) are
similar to those at the edges y = 0 and y = b. They are
w(O,y1T) = w(a,y,T) = 0
(21)
Again the rotation, in this case ~' has been specified as zero
rather than M:xy· Substitution of the expression for w, Qx, and Qy
from equation (20) into the boundary conditions (eq. (21)) results in
6 homogeneous equations in the 6 unknown coefficients Bi. In order
for a nontrivial solution to exist, the determinant of the coefficients
of Bi must vanish or
+ r-~ l ~l 1 - 13i 131 + r-~ l l 1 - 13~ 132
+ r-~ l j l 1 - 13~ 133
+r -[ l j l 1 - 13~ 134
0 0
~ 1 ~~ [ l J~ [ l + J~' [ l +~~ 0 0 +r- +r-1 - 13i 131 1 - a~ 132 1 - a~ 133 1 - 13~ 134
-131 -132 -133 -134 rl35 rj36 2 2 1 - 13~ 1 - 13~ 1 - 131 1 - 132
= 0
-13 em,,_ ~ m ~ I I\)
-133 e 3 0 -132 e -134 e rl35 e~ rl3 e '11l6
• 1 2 1 - 13~ 1 - 13~ 1 - 13~ 6 1 - 131
1 1 1 1 - - - - .135 136 131 132 133 134
m1 eme em3 emJ+ a eTI!Jj 136 emf, .!t__
131 132 133 134 5
where lilt 131 = -.
1(8
- 21-
The sixth order determinant '11JS.Y be reduced to the following
fourth order determinant
1 1 1
m3 e
1
= 0 (23)
Equation (23) is the same as that obtained for a panel rigid in shear
(r=O) except that the values of the ~ are different for a panel
with finite shear stiffness (see eq. (17)~ It should be noted that
the solution is independent of ~ and mo· Equation (23) can be
reduced to
(~ - mi)(~ - ~) sinh (mi ~ m~) sinh ("2 ~ mi.)
+ (m~ - mi) ( ~ -~ sinh ("I ~ "2) sinh ( m~ ~ IDJ,) :O (24)
It will be remembered that the roots mi_ appearing in equation (24)
are solutions of equation (17) which is a fourth degree equation. The
form of these roots for r = 0 is known from previous analysis, and
this general form will be extended for the present problem. The
roots will be ta.ken as
- 22 -
~ = 5 + a - i5 = ~ - i5
m3 = s - a + € = p + €
Init = g - a, - € = p - €
When r = o, g = 0 and the roots assume the form found for a
panel rigid in shear. The term s appears in each root since the
sum of the roots must add up to the negative of the coefficient of
m3 in equation (17). Additional useful relationships between the
(25)
coefficients in equation (17) and the roots mi can also be obtained
from the rules governing the roots of algebraic equations. These
relationships are
52 = ">-. + .,(2. A + a.2 _ 352 _ gi_3+ 1(2 Xs 4a. 2 a, 2a,
(26)
(27}
ji" = -?+ rr CL2 - e2) (CL2 + 52) + !4 + t2 (82 - e2 - 2CL~ -2a.t (·2 + 8~ (28)
Substitution of equation (25) into equation (24) results in
u.2 + 52)2 + 4CL2 (82 - ,2) + 4t2 (4a.2 + &2 - .2) J sin 8 sinh €
- 8e5 (a.2 - g2)(cosh € cos 5 - cosh 2a.) = O (29}
- 23 -
When ~ = O, equation (29) is identical to that obtained by Hedgepeth
in reference 1. For a given panel and loading condition, combinations
of airf'l.ow (A) and frequency (ro) which satisfy equation (29) represent
a possible motion of the panel. In fl.utter analysis, if A is plotted
against frequency, two modes will coalesce and any increase in A
results in a complex value of a:f!.. Thus oo is the square root of a
complex number and one root must have a negative imaginary part.
Examination of equation (15) reveals that divergent or flutter motion
would occur if the imaginary pa.rt of c.u is negative. Thus in order
to determine a fl.utter point, it is necessary to obtain A as a
:f'unction of frequency in order to find the value of A for which the
frequency becomes complex, which occurs at the maximum point in a
frequency loop.
Since frequency and the stress resultant Ny appear in the
analysis only in the parameter q>, it can be seen that plots of A
versus frequency :for different values of Ny would only involve a
shift in the frequency axis and the critical value of A would
remain unchanged. Hence Acr is independent of Ny- just as was
found in reference 1 for the plate rigid in shear. Thus fl.utter
solutions can be found from plotting A against q>. SUch plots can
be obtained as follows.
For a given panel and loading, e, kx' and r are specified.
For a given q>, A is varied until equation (29) is satisfied in
order to obtain one point on the plot. However, the quantities
- 24 -
appearing in equation (29) cannot be determined directly unless a
is known. Elimination of 52 and e2 in equation (28) by the use
of equations (26) and (27) results in the following cubic equation
for a2
(30)
where
The procedure for the solution is now as follows. Select 8, kx, r,
and cp. Guess at a trial value of ""}:. Now all the quantities
necessary to calculate Dl' n2, and n3 are known and a is
determined from equation (30), 5 and E are determined from
equations (26) and (27). These values a.re substituted into equation
(29). If this set of values does not satis:t'y equation (29) a new
value of ~ is selected, the process repeated until a solution is
found. Next a different value of cp is selected and the corresponding
value of 7' determined. These results can be plotted as shown in
f'igure 2 and the critical value of 7' is at the maximum of the
curve. The curve intersects the 7' equal zero line at the panel
natural frequencies which can be obtained directly from equation (17)
- 25 -
~I
Figure 2.- Schematic diagram of frequency loop. Dynamic pressure parame-eter ~ versus frequency para.meter ~·
- 26 -
and (24) and noting that £ == 'A == O.
(31)
where n refers to the nth mode in the flow direction.
Preflutter solution.- Movchan, in reference 2, pointed out that
a very simple solution satis:fied the stability equation for panels
rigid in shear, namely o = 2nn: and e = 2a.. This solution which
gives a simple algebraic expression for 'A falls on a CUI"V'e of 'A
against frequency though of course not necessarily at or near the
top of the loop, hence has been described as the preflutter solution.
However, calculations for panels rigid in shear show the prefiutter
solution to be very close to the critical value of 'A as A
increases in the negative direction. Because the exact solution of
the transcendental equation becomes more difficult in this range, the
preflutter solution is very useful. Examination of equation (29)
reveals that taking o = 2nn: and e = 2a. will also be a solution
for a panel with finite shear stiffness. The value of 'A that
results from this solution can be found as follows. The value of
a.2 can be obtained from equations (26) and (27) as
a,2 = -
From the same equation 'A can then be determined as
(32)
- 27 -
(33)
When r is zero, £ is zero, and !\ = A so that equation .(33) is
the preflutter solution found by Movchan. The integer n is found
to indicate the frequency loop, and it can be seen that n = 1 gives
the lowest value of A; hence it is the most critical. When r is
not zero, the terms involving g make the solution more difficult.
However, g and I\ a.re related by
~ = 5'1 + r) 4~ e2 = 4~rg r
(34)
Using equation (34), equation (33) can be solved for ~ as
(4 - X)(A-- 60) + r2 + I'A ±
(;)2 = 12
r(4 - A)(A - 6o) + r2 + rX\2 _ ~(4 + r)(10 - A) 2 (4 - A)
\ 12 J 27
8(4 + T') (35)
Inasmuch as equation (35) involves A which contains the frequency para.meter q>, a trial and
error solution is still involved. First a value A and r are selected and B is calculated
from equation (28) usill8 equation (32) for a, equation (35) for g, and noting that € = 2a.
Next q> is eliminated from the definition of A and B given in equation (17) and the resulting
expression solved for k:x to give
2 + r a2 - :w +A_ rD kx = 1 + r 92 1 + r)
( 1 + r) a2 + r A _ rB ) ( 36) e2 (1 + r)
If' kx calculated from equation (36) is not the value desired, a new value o:f A can be
selected and the :process repeated until the desired value of' ~ is obtained. It should be
noted that equation (35) ha.a two roots and either may apply depending on the value o:f kx·
~
- 29 -
Once a value of A is found that results in the desired k,c, the
calculation is essentially complete since A may be obtained from
the definition of ~ as
(37)
Preflutter solutions do not exist for all combinations of loading and
panel proportions but this solution always exists and can be very
useful for large negative values of A.
Flutter mode shape.- Once a solution has been obtained for
Acr' the flutter mode shape can be determined from equation (22)
though of course the amplitude is indeterminant. The solution for w
is
w = Re sin ~ ei<IYT e ! ~[(Te-a sinh • + Ve!" sin o)( ~cos 5 ! - e-a ~ cosh ::) + (se-a sinh € - Vea cos o + e-a V cosh €
- ue-a sinh €)ea i sin o ~+(Te-a cosh € + uea sin o
- Te" cos 5 - Se" sin 5) e-" ~ sinh < ~ J where
2c sin 2d T = ----------------------------------~
s
(1 + r)2 + r2 c2 - 2(1 + r)rc cos 2d
2(1 + r + rc2 - 2rc cos 2d - c cos 2d)
(1 + r)2 + r2 c2 - 2(1 + r)rc cos 2d
(38)
- 30 -
c =---1(2 92
Clamped leading and trailing edges .-For clamped supports the
deflection and the rotations in both directions are zero. Thus the
appropriate boundary conditions from equation 12 can be written as
w(O,y,T) = w(a,y,T) = 0
(J>x (o,y,T) = q>x: (a,y,T) = o (39)
Again six homogeneous equations in the coefficients Bi are obtained
by substituting equations (20) into the boundary conditions (39).
Solutions of this set of equations are only possible if the determinant
of the coefficients vanish. After some simplification, this deter-
minantal equation can be written as follows.
G l + j L 1 - 13~ 131 [ l +J~ 1 - 13~ 132 [ l +r]i
1 - 13~ 133 [ l + r] l
1 - 13~ 134 0 0
l J e"'L G l J .m2 [ l J 0m3 [ l J em4 -+r - +r - +r - +r - 0 0 1 - 13f 131 1 - 13~ 132 1 - 13~ 133 1 - 13~ 134
1 1 1 1 -r -r 1 - 132 1 - 132 1 - 132 1 - 132
1 2 3 4 I:: 0
em1 em2 em3 eID4 -re~ -rem6 I (40)
1 - 132 1 - 132 1 - 132 1 - 13~ \..N I-' 1 2 3
..1.. ..1.. .1.. 1 135 136 131 132 133 134
ml e~ m3 m4 135 e~ 136 e il6
e e e - - - -131 132 133 134
Expanding equation ( 40) results in the followir..g:
•1 •2 sinh ":i_ - "2 [ ~ - ) m3 -m4 )( m3 +m4 p3 p4 p§ sinh ~ sinh 2 + ~5 (P3 - ;34 cosh ? - cosh ~
+ •1 v3 sinh m3 - ml Ii( - I 1 -L
2\ ~-m4 ;32 f34 ;35) sinh m5 sir..h 2 · + 135
(, \ ( !12 + ID.1+ \32 - ;34) cosh 2 - cosh m5
cosh m, ~ m4 )]
cosh "2 = ~]
+ ti t 4 sinh ml. : mi{ (i - ~2 33 a~) sinh "5 sinh "2 ; m3 + .a5 02 - a3) ~osh "2 ; m3 - cosh "5 cosh "2 ; m3) J \;;
+ V V sinh m2 - ~lt(1 - j3 ~ 132) 23 2 ~ 145
II)_ - m4 ( ) fr II)_ + m4 sinh m5 sinh 2 + ;35 ;31 - ;34 \osh 2 - cosh ~ cosh ":i. ~ m4) J
m4 - ~Ii( ~ + 1'2 V4 sinh 2 ~1 - i31 i33 i3~ sinh "5 sinh ":i_ ~ m3 + a5 ~l - 3~ (cosh ":i_ ; m3 - cosh "5 cosh ":i_ ~ m~J
m3 - m4 ll( 2\ + v3 v4 sinh ? ~ 1 - i31 i32 135) sinh ~ ~-~ )( ~+~
sinh 2 + ;35 ~l - (32 \osh 2 - cosh ~
= 0 (41)
where ti = 1 + r - rPi
substitution of the expression for the mi from equation (25) into equation (41) yields the
following:
5P5 sinh € (cosh Tl -rc0
ff(4a,2 + €2 + 52\(1 (p2 + €~\ COB 8 COSh ~) ~ rc2 a2 -; r + l - rc2 a2 J
8e2 J - rc4 a~
EP fi(1ia.2 _ <2 _ ry f1 ~2 - s2\ ~ l +~sin 5 (cosh p - cosh e cosh ~) ~ rc2 92 \[ + 1 - rc2 92/ - rc4 a4 ~
5€ [ 4 T}P 2 ( 1 T}2 + 5~ ( 1 p2 - Eaj + 2 _ sinh ~ (cosh 2a. - cosh e cos 5) rc2 92 - P5 \; + 1 + rc2 92 -}~ + 1 + rc2 92 ~
~ fi l 4a.(1 \ 211(P2 - e2) 2P(1l2 + 52)1 + rc2 02 \sinh p cos 5 - sinh Tl cosh € + sinh 2a. cosh ~ lrca \f' + l} + rc3 03 - JC3a3 J
~52 (4a.2 + 52 - e2) [l ~ ~ (p2 - e~ (112 + 52) {l ~ PT} ~ + sinh € sin 5 sinh ~ - 2 2 - + 1 + 4 4 + - + 1
2 JC a r re a r 1(2 a2
-2(;+~ 2 e2 52 ( 82 + e2 + P2 _ 112) 2 2( 52 P2 _ e2 11~ 135 -- - + -------
1(4 04 2rc4 94 rc4 94 = 0 (42)
\>I \>I
- 34 -
As 8 approaches zero in equation (42) the results approach that for
a clamped beam, which is
~ l q2 + 52)(1 p2- e2) 20€ ~ + rr2 ~ + 1{2 (cosh 2a - cosh € cos a)
{ (4a.2 + 52 _ .2)( i_ + g_ .ll.!l + (n2 + 5~02 - ·~) (43) r'2 r' rr2 rr
- .!!:._ E2 52 J sin o sinh € = 0 r' rr2
where r' = 1C2 D
a2 D Q.
As r approaches zero, equation (42) reduces to
2BE(cosh 2a - cosh € cos B) + (4a2 + 52 - €~ sin o sinh E = 0 (44)
which is equivalent to the solution obtained by Houbolt in reference
8 for flutter of a clamped panel rigid in shear.
The no flow frequencies can be obtained from equation (42) by
noting ~n equation (17) that for ~ = A= o, o and E are then
given as
5 ~ .f + ~2 + 4B
€ = ,fl. +~2 +liB
with a= ~ = p = 0
- 35 -
Equation (42) then becomes
- cos
€13 - ~ sin o(l - cosh e:
rc8
135 0€ (1 - cosh
1(2 92 ~ 52 ~(l €2) € cos o) sinh ~ - + 1 + -- - + 1 -r rc2- 82 r rc2 92
(46)
132 + .:.2. sinh e: sin o
2
-(5~ \ ·2{2]= 0 sinh~ -- -+1--- -- -+l+--~ 52 (1 €2 ~ €2 (1 52 )
1C2 82 r rc2 92 rc2 92 r rc2 92
213 5 1( 0
Equation (46) may be factored to give two frequency equations
corresponding to symmetricaJ. and antisymmetricaJ. modes.
52 + €2 ---- tanh ~ == 0
2
symmetricaJ. modes
-+1--- 0 ( 1 e:2 ) r rc2 92
antisymmetrical. modes
€ coth -2
(47)
(48)
- 36 -
Equation (47) is equivalent to that obtained by Seide in reference 9
for the compressive buckling of a sandwich plate clamped on the unloaded
edges. Solutions for buckling are obtained from the present analysis
by setting the frequency w equal to zero and solving for combina-
tions of Nx and Ny that satisfy the frequency equation.
Numerical results for flutter are obtained for the clamped panel
or beam in the same manner as for the simply supported panel except
equation (42), (43), or (44) is used instead of equation (29).
- 37 -
VII . RESULTS AND DISCUSSION
Simply Supported Leading and Trailing Edges
Values of Acr have been determined for the complete range of a b
from zero to infinity for several values of the shear stiffness
parameter r ranging from zero to two, and these results are shown
in figure 3 for kx: = O. Calculations were also made using the
preflutter solution. a At the larger values of - the value of A b
calculated from the preflutter solution virtually coincided with the
maximum value of A on the frequency loop. In as much as calculation a of the frequency loop is very difficult for larger values of - the b
preflutter value of A was used in extending the curves to a equal b
infinity. In order to limit the range of numbers to reasonable values,
the cube root of the flutter para.meter is plotted and the results are
presented in terms of para.meters that are based on the shortest
dimension of the panel. Thus when ~ < 1 the para.meter , n:2D r = :::-2'T)
a DQ
and are introduced. When r = o, the shear stiffness
is infinite and the results are the same as obtained in previous
investigations (refs. 1, 2, and 8). As would be expected, decreasing
the shear stiffness (r increasing from zero) results in a lower value
of the dynamic pressure at flutter. When the results are presented
as in figure 3, it can be seen that flutter is essentially independent a of length for b > 10 and is essentially independent of width for
a b < 0.25. These ranges tend to increase as r increases. The
l' 1/3
- 38 -
9·~~~~~~~~~~~~~~~-,~~~~~~~~~~~~~~~9
l-l-+--+---+---+-+-<-+---+--+--i-~-+---l--l-l-l---1--+---11-+-+---l--l-~-4-+-+--1--1-l--4-+-l---1--1-1--4---l--.I
~-+-+--j--j-+-+-+---+-+----+---+--1 I I I I I I I _,_,_ _ __,___.____,___~__._,__,__,____,__,___,___,_____,___.____,__...._.._.
t--+--+-+--+--+-f---+--+-+--+--J--+1- r o r r ' = 0 _,___..__.___,_____,____,___,_~, _ _,__,_ _ _.___,__~-+--+--+-+--+-1 8 I I 8
1--+-+---+--+-+--l--+---+---+---+--+-+-+----t----t----t--lc..-!'t......r~ !'\.~-l--l-l-4---+-+--l-+-+--+---+-4-1--4---1--l--l--l-I I--+---+---+--+-+-•-+-~ ~ L+-1-+-=l-""i..--F--+--+--'. '6.0..t?.--!'l.+-+---+----1-+--+-+-+---+----l--l--+-++---+----1-+--+-1
i....-- i..-- I'\ \
t--+-t--+--+--J--+-+--+--+--+--+----j-+--+--l-1---1----~· - -+-+--l-+-l-l--+-+--+-~l-l--+-+---1--1-1-1-+--I
2 t--+--t--t--1-4-+-+--+--+- +---+--+--+-!' - j----+--+---+---+-t-+--+-~-+-l--l-!-l--+-+-!-+-+-+-+--+ l-+--4--1--+--+--l!-l--, -+--ll--l--1-1--1--+-+-+-+-+-+-l--+-+-l-l 2 1---+---1--+--4--+- ·t -·-~ -+- - - --r--+----1--1___.__,__- --t-r-~~ -- . >-+---+---+---+-- _ _J__L_ --~ ---~ _ ----+--1-i--4----+---'--·"- -1---+--1-!---+---+---+--+-<---+---+---+--1---,_._,__--+---+--+---l-~- --1----jl-+--+---+- ---t - ;_I--. - - -l--1--+--'---'--~--'--
J---1-+--+---+ -l--+-+---1-+- . -•--+--+-+--1--1-1-1-+--I
-r--+-- i--1- <-----.---+---+--l-+--4-+--1--1-J--+
I
0 .2 .4 .6 .8 I. 0 .8 . 6 .4 .2 0 a k o a
Figure 3.- Flutter values of dynamic pressure. Simply supported panel, kx = o.
- 39 -
values of r and r' shown correspond to a wide range of shear
stiffness that might occur in practice and the results show that
flutter can occur at dynamic pressures considerably lower than that
of a panel rigid in shear.
A more detailed study of the effect of shear stiffness on the
flutter behavior of sandwich panels can be obtained by examining the
frequency loops as indicated in figure 4. These results are for an
unstressed square panel with several different shear stiffnesses.
The figure shows that decreasing the shear stiffness causes a decrease
in Acr and also causes a decrease in flutter frequency and the
natural frequencies of the panel. (The natural frequencies are given
by equation (31)). For larger values of r, the curves have a
tendency to lean over to the right. The lowest flutter point occurs
with the coalescence of the two lowest frequencies. This fact was
found to be true for all proportions and all values of r.
The effect of shear stiffness on flutter mode shape is shown in
figure 5 for the sam.e panels discussed in figure 4. The mode shapes
were determined from equation (38) for A equal. Acr· The longitudinal
mode shapes that occur at a cross section y equal constant and at an
arbitrary time are shown. They have been normalized to the same
amplitude. The effect of decreasing shear stiffness is to concentrate
the deflection toward the trailing edge of the panel.
Clamped Leading and Trailing Edges
A limited a.mount of calculations were made for clamped panels
and the results shown in figures 6 and 7, In figure 6 Acr is
- 4o -
600
r = 0
500
400
l 300
200
100
0 5 10 15 20 25
Figure 4.- Frequency loops for simply supported panels.
w . 2
- 41 -
.4
--:>-... , / , '\
/ / \ \ r = .4/ 2/ \\
/ ,' \\ / / ,\\ ,, .,,,, \
~ ,,, ....... --· ___ ........ -
x a
.6 • 8 1.0
Figure 5.- Mode shape at flutter for an unstressed square sandwich panel. a/b = l, kx = O.
10
8
6
A.' 1/3
4
2
0
10
8
6
A.' 1/3
4
2
0
- 42 -
--- -- ---
• 5 1.0 rr
. 5 I. 0 r I
CI amped Simple support
-----------
I. 5 2.0
--------
I • 5 2.0
Figure 6.- Effect of shear stiffness on flutter of sandwich panels.
- 43 -
600 Clamped
---- Si mp I e support
500
400
A 300
200
100 I I I I I
' I
0 10 20 30 40 ~
Figure 7.- Effect of shear stiffness on frequency loops, clamped and simply supported panels. a/b = 1, k;x = 0.
plotted against r'
- 44 -
for an unstressed panel with a equal 0 (beam) b
and ~ equal 1. Also shown for comparison are the corresponding b
results for a simply supported panel. At r equal zero the values
agree "With those established by previous investigators. As r
increases one might expect the curves for the clamped panel to approach
the curves for the simply supported panel just as the buckling loads
and natural frequencies approach each other for r greater than o.4.
Instead, however, the curve for the clamped panel is below the curve
for the simply supported panel at the larger values of r. The
reason for this is not known since one normally expects boundary
restraints to increase a panel's resistance to flutter. A similar
situation was found for simply supported panels under tensile loadings.
For certain ranges of para.meters an increase in shear stiffness causes
a decrease in Acr· This phenomena (an increase of panel stiffness or
boundary restraint resulting in a lower value of Acr) may actually be
a correct picture of the panel behavior or it may represent a range
of shear stiffness and loadings for which the theory used does not
properly describe the panel response in the presence of air forces.
A better insight into the flutter behavior is obtained by looking at
the frequency loops for the two boundary conditions. Frequency loops
corresponding to an unstressed square panel are shown in figure 7 for
r = 0.05, and o.4. The curves intersect the A = O a.xis at values
of~ that can be obtained from equation (47) and (48) for clamped ends.
As A increases the first and second modes coalesce. For r = 0.05
- 45 -
the clamped panel has the higher value of Acri for r = o.4 the
simply supported panel has the higher value of f..cr• a As b increases one would expect the effect of the boundary
conditions at the leading and trailing edges to disappear. That this
is the case is illustrated in figure 8 where ( t..cJ l/3 or0~r) l/3 is
plotted against !: for a few values of r or r' • At higher values b
of ~ the results for simply supported ends merge into the results b
for clamped ends. As noted from the previous figures, the results
for the clamped ends are lower than for the simply supported ends
when r > 0.2 a and b is small.
Comparison with Two-Mode Solution
In reference 5, McElm.an presented a two-mode Galerkin solution
for the flutter of a simply supported sandwich panel using the same
basic equations as used herein. It is known that for r equaJ. zero
the two-mode solution diverges from the exact solution as A increases
in the negative direction and can be considerably in error although
the trends are correct. A comparison between the exact solution and
the two-mode solution is shown in figure 9 where Acr obtained from
both solutions is plotted against r a for two values of b. An
expression for Acr for the two-mode solution given in reference 5
can be written in the notation of the present paper as follows
1 - kx + ~------------------------------------------~
a4 r + ------------------------------------------
4 r + 5a2 (1 + r) + a4 (2 + r)
(49)
10
6
l' 1/3
or l 1/3
4
2
0 2
- 46 -
4 a b
Clamped Simple Support
r or r'= 0
.2
2
6 8
Figure 8.- Variation of flutter para.meter with a/b for clamped and simply supported panels.
- 47 -
9.....-r-r-o-.--r---r--,---,-r~r""T-r-~I-,-l..,......,_,..-,--r--r-rl~I-.-...,.-,.......,...---.--.--.-.---.-..-.--.--.-.......... -.-. - - -- ·---~- !---; __ -~-~~ -111- --
- ·--1--~ --
J --~~-1 ---l - -- - - - -: I
- -1--.___ r !-- - - ...____ --
- - - - Gate r kin, (Ref. 5)
--- Exact
-L-
two mode -'-- .__
-+-+---+-- .._-+---- L- L-
7 r\':-t--+-~..,...:+-+-+-+-t-11-+-+-+-+-+-+--+--+-+-+-+-+-+-i-+-+--+-+-+-+-+--!--+-.+-+--+--1!-4~......j.~ \ '\
+-i----1--<-·+-- ------ ._ -- - --1---~ - ..____ --
-+--+--+--· --1----1------
··'-- -r· --. . -·
- ---l--1-------l-.-
'- I ---...L_ ~t--ll--l-+-+-..i-~-+-~-l--+--l--+!-+-+---l--+--1----l-- ·t·--1--+--+; ~~-1.d---~,:...:.::.t--ll--l--+--- '--· 5 t-t--t--t-+-+-+,.,._+-+-+-+-+-+-+--+-+-+-+--+-+__,,-+4-+-1--F==i"" ...... ~-+-+-1-1--+-+-l-l--+-I
" I I "' _,_._ __ -.... ....._ I /3 ' - -- - ·- __ ,___ - ,_ j r "'- r--..,.....=·--•i.;,....=-~- -A. t--1--11--l-+--+--+-+--1--+"~•,-+-+-+--+--1if--l-+-+-+-+--1- ~ ~~- ~-~ . - -·· ·-"""'
t--t--ll--l-+--+--+-+--l--+--l-~..+~-+-+--+--1---l--+l. _...__,__+--+-1--+-+--'l.--l-
-l--+--.+--l-..... -+. ...... -+-..... -+,_--f-· --1-~-- --- - I) ~ = I. 0 4
t:::~-~""""'--t---t---+- .... .... / ;..............., ....___ -- --- -~ -- ---- ...... i.-~ - ... """_ - /-
t--H-t-t-'F"-"'~,__::+_-1_--+-+--+-+- - -. - --
_i__ - - ~I. _L __ --- -
- ~---,;..;:::..:._. -.. __ ·----- -- ~ --~--
3 ~'~~~~~~~~~~~~~~~~~~~~~~~~~~~t-~-~·~-~~~~~~~~~-~~~~--1~--~-~~-~~ '
t--r-r--1-;--r--+0-·+~~----=-c+-~~--+--+-+-:1-1 -+--+--l-+-+-+--+-~--+-+-P=~- %= 5.0 2 r- -+--;_ /
t--t-l--l--+-+-+-+-l---l--l--l--+-+-+-~1--J--!-+-+-l--l-_-l-_-1--4._-+ __ -+--+""t-~*"--+--L-~db-l-l---l---+---l---I - -- -i.- ._ -
0 • 2 .4 • 6 .8 1.0 r
1.2 1.4 1.6 1.8 2.0
Figure 9.- Comparison of fl.utter boundaries from two-mode GaJ..erkin solu-tion and exact solution. ~ = O.
- 48 -
The f'igure shows that for r constant, the error in the two-mode
solution increases as a increases which would be expected based on b
the results for r equal zero. The figure also shows the error to
increase with r for a constant. b
The range of r and ~ for which b
the two-mode solution gives reasonably accurate results is quite
small.
- 49 -
VIII. CONCLUDING REMARKS
The flutter of an isotropic sandwich panel subjected to the air
forces given by two-d.im.ensionaJ. static aerodynamics has been considered.
The sandwich theory of Libove and Batdorf which allows transverse shear
deflections was used and an exact solution to the problem was obtained.
From the results of the analyses the following conclusions can be
made concerning flutter of sandwich panels:
1. Flutter is not affected by inplane loadings perpendicular to
the direction of air flow.
2. For unstressed panels, decreasing the shear stiffness causes
a decrease in flutter dynamic pressure.
3. For panels without compressive stresses in flow direction,
flutter is essentially independent of length for a greater than b 10 and essentially independent of width for a less than 0.25.
b 4. For large values of shear stiffness, panels with clamped
leading and trailing edges are more flutter resistant than simply
supported panels; however, at smaller values of shear stiffness the
analysis shows the clamped panel may actually flutter at a lower
dynamic pressure than an identical simply supported panel. As a b
increases, the results for either boundary condition are essentially
identical.
5. A two-mode Galerkin solution to the problem becomes increas-a
ingly inaccurate as b or r increases.
The results of the analysis indicate that more theoretica1 work
is necessary in order to understand the role of boundary conditions
- 50 -
in flutter of sandwich panels. In addition experimental work is
needed to identi:f'y the practical problems that may a.rise in designing
sandwich panels to resist flutter. However the numerical results of
this paper should provide a good basis for evaluation of an experimental
flutter program for sandwich panels.
- 51 -
IX. SUMMARY
An exact solution to the problem of flutter of a sandwich panel
has been obtained. The aerodynamic loading is assumed given by two-
dimensiona.l static aerodynamics. The sandwich plate is considered to
be isotropic but to have a finite transverse shearing stiffness. The
strea.mwise edges are simply supported and the leading and trailing
edges may be clamped or simply supported.
The results of the analysis are presented in graphical form in
plots showing the flutter dynamic pressure as a f'unction of length-
width ratio for various values of shear stiffness. Results are
presented for length-width ratios from zero to infinity for both
clamped and simply supported leading and trailing edges. Comparison
with a previous two-mode Galerkin solution to the problem is also
given.
- 52 -
X. ACKNOWLEDGMENTS
The author would like to express his thanks to the National
Aeronautics and Space Administration for allowing this work to be
done as part of a research iIIVestigation being carried out at the
Langley Research Center. Special thanks a.re due Mr. Larry L.
Erickson for programing the rather long and complicated equations
which were used in obtaining numerical results. And finally,
thanks a.re due Professor Daniel Frederick of the Virginia Polytechnic
Institute for his comments and suggestions during the course of this
iIIVestigation.
- 53 -
XI. BIBLIOGRAPHY
1. Hedgepeth, John M.: Flutter of Rectangular Simply Supported
Panels at High Supersonic Speeds. Journal of Aerospace Sciences,
Vol. 24, No. 8, August 1957, pp. 563-573, 586.
2. Movchan, A. A. : On the Stability of a Panel Moving in a Gas .
NASA RE ll-21-58W, 1959·
3. Cunningham, H. J.: Flutter Analysis of Flat Rectangular Panels
Based on Three-Dimensional Supersonic Potential Flow. AIAA
Journal, Vol. 1, No. 8, August 1963, pp. 1795-1801.
4. Bohon, Herman L., and Dixon, Sidney C.: Some Recent Developments
in Flutter of F1at Panels. Journal of Aircraft, Vol. I,
No. 5, September-October 1964.
5. McElman, John A.: Flutter of Curved and Flat Sandwich Panels
Subjected to Supersonic Flow. NASA TN D-2192, 1964.
6. Libove, Charles, and Batdorf, S. B.: A General SmaJ.l Deflection
Theory for Flat Sandwich Plates. NACA Rep. 899, 1948.
7. Dommasch, Daniel 0., Sherby, Sidney S., and Conna.lly, Thomas T.:
Airplane Aerodynamics. Second ed., Pittman Publishing Corp.,
1957, pp. 127-131.
8. Houbol t, John C. : A Study of Several Aerothermoelasti c Problems
of Aircraft Structures in High-Speed Flight. Nr. 5
Mitteilunger aus dem Institut f'ur Flugzu.rystatik und Leichtbau.
Leeman (Zurich) c. 1958.
- 54 -
9. Seide, Paul: Compressive Buckling of Flat Rectangular Metalite
Type Sandwich Plates with Simply Supported Loaded Edges and
Clamped Unloaded Edges. NACA TN 2637, 1952.
The vita has been removed from the scanned document
Fl.UTI'ER OF SANDWICH PANELS AT SUPERSONIC SPEEDS
By
Melvin S. Anderson
ABSTRACT
Panel flutter is an important design consideration for vehicles
traveling at supersonic speeds. Most theoretical analyses of panel
flutter consider the motion of the panel to be described adequately
by classical thin plate theory. In such a theory, transverse shear
deformations are neglected which is a reasonable assumption for solid
plates. For a sandwich panel, neglect of transverse shear deforma-
tions may not be a good assumption in flutter analysis inasmuch as
studies have indicated that the vibration and buckling behavior of
such panels can be affected significantly by shear deformations. An
analysis which considers transverse shear deformations is presented
in order to determine the effect of finite transverse shear stiffness
on the flutter behavior of sandwich plates.
The sandwich theory used is due to Libove and Batdorf. The
essential feature of this theory is that straight line elements
perpendicular to the undeformed middle surface remain straight and
of the same length but are not necessarily perpendicular to the
deformed middle surface. The aerodynamic loading on the panel is
given by two-dimensional static aerodynamics. The adequacy of such
an approximation has been demonstrated for panels rigid in shear and
the mathematical simplicity allows closed-form solutions to be found.
The analysis proceeds from consideration of the equilibrium of an
infinitesimal element. If equations are written in terms of the
deflection and two shear deformations for equilibrium of forces in
the z direction and equilibrium. of moments about the x and y
a.xis, three differential. equations involving the three unknown
displacements are obtained. This system of equations is of sixth
order with constant coefficients, but for simple support boundary
conditions on the stream.wise edges an exact solution can be obtained.
The associated characteristics equation can be factored into a
fourth degree equation and a second degree equation; thus an
analytical expression can be obtained for the characteristic roots.
The solution just described is a general solution for the
motion of a sandwich panel simply supported along stream.wise edges
and subject to inertia loading and aerodynamic forces given by two-
dimensional static aerodynamics. Any combination of boundary condi-
tions consistent with the sandwich plate theory used can be applied
at the leading and trailing edges. Two cases are considered: simply
supported leading and trailing edges and clamped leading and trailing
edges. With the use of either set of boundary condition, a transcen-
dental equation is obtained which is satisfied by various combinations
of frequency and dynamic pressure. The dynamic pressure necessary to
cause the frequency to become complex corresponds to divergent
oscillatory motion or flutter.
Values of the flutter dynamic pressure have been calculated as a
function of length-width ratio for a large range of shear stiffness.
For inf'inite shear stiffness the results agree with those established
by previous investigators. As shear stiffness decreases, the flutter
dynamic pressure usually decreases also. An unusual result of the
analysis is that at low length-width ratios, a clamped panel has a
lower flutter dynamic pressure than a simply supported panel even
though the vibration frequencies are higher for the clamped panel.
Results are not presented for panels with norm.al inple.ne loadings
but they can be obtained from the equations given. The analysis
shows that flutter is independent of normal inplane loadings perpen-
dicular to the flow direction just as was found for panels rigid in
shear.
An approximate two-mode Galerkin solution to the problem has
been obtained by a previous investigator. Comparison of the exact
solution to the approximate solution shows the approximate analysis
to be in increasing error as length-width ratio increases or shear
stiffness decreases.
Recommended