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.