AEROELASTIC PROPERTIES OF STRAIGHT AND
FORWARD SWEPT GRAPHITE/EPOXY WINGS
by
BRIAN JEROME LANDSBERGER
B.S., United States Air Force Academy
(1972)
SUBMITTED TO THE DEPARTMENT OF
AERONAUTICS AND ASTRONAUTICS IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
MASTER OF SCIENCE
IN
AERONAUTICS AND ASTRONAUTICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February,
1983
Massachusetts Institute of Technology 1983
Signature of
Author
Department of Aeronautics
nd'Astronautics
December 14, 1982
Certified
by
John 'Duqundji'
Thesis Supervisor
Accepted
by
ArchiveSChairman,
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR 3 0 1983
.
r%
it
Harold Y. Wachman
i i
Departmental Graduate Comittee
AEROELASTIC PROPERTIES OF STRAIGHT AND
FORWARD SWEPT GRAPHITE/EPOXY WINGS
by
BRIAN JEROME LANDSBERGER
Submitted
to the Department of Aeronautics
14,
in
1982,
partial
fulfillment of
the
and Astronautics
requirements
on December
for the degree
of
Master of Science in Aeronautics and Astronautics.
ABSTRACT
The aeroelastic deformation, divergence and flutter behavior of
rectangular, graphite/epoxy, cantilevered plate type wings at zero sweep
and thirty degrees of forward sweep is investigated for incompressible
flow. Since the wings have varying amounts of bending stiffness, torsion
stiffness and bending-torsion stiffness coupling, they each have unique
aeroelastic properties. A five mode Rayleigh-Ritz formulation is used to
calculate the equation of motion. From this equation static deflection,
steady airload deflection, divergence velocities, natural frequencies and
flutter velocities are calculated. Experimental two dimensional lift and
drag curve data and approximations to three dimensional aerodynamics are
used to calculate the aerodynamic forces for the steady airload
analysis.
The Weissinger L-Method for three dimensional aerodynamic
forces is used in the divergence analysis.
The V-g method is used to
make flutter and natural frequency calculations.
Tests on a static
loading apparatus gave static deflections, while wind tunnel tests gave
steady airload deflections for the wings at zero sweep, and divergence
and flutter behavior data for all wings at both zero sweep and thirty
degrees forward sweep.
Wings were tested from zero to twenty degrees
angle of attack for airspeeds up to divergence, flutter or the thirty
meter per second limit of the tunnel.
Static deflection, natural frequencies, steady airload, divergence
and flutter for the straight wing were predicted reasonably well by the
theoretical calculations.
For the swept forward wings, calculated
flutter speeds were beyond the wind tunnel capabilities, while calculated
divergence speeds were reasonable when divergence did occur. When swept
forward, before reaching predicted divergence
speeds some
lightly
bending-torsion stiffness coupled wings went into a torsional flutter,
characterized by a large average bend which caused a high wing tip angle
of attack.
This flutter was not predicted by the theory used.
The
different ' wings
exhibited
markedly
different
stall
flutter
characteristics.
Thesis Supervisor: John Dugundji
Title: Professor of Aeronautics and Astronautics
ACKNOWLEDGEMENTS
I thank Professor John Dugundji for his continuous guidance during
all phases of this project.
I also thank Rob Dare, a fellow student, who
assisted me during all experimental work.
Finally,
to
all
the
staff
of
M.I.T.
who's
instruction
and
assistance made this project possible, and to the United States Air Force
which
funded
my
hearty thank you.
education
and
the
contract
covering
this
project,
a
TABLE OF CONTENTS
Page
List of Figures..................................................6
List of Tables..........................................................7
List of Symbols..........................................................8
Chapter
1. Introduction........................................................
11
2. Theory
2.1
Rayleigh-Ritz Analysis...................................12
2.2
Static Deflection Analysis.................................23
2.3
Steady Airload Analysis...................................25
2.4
Divergence Analysis..................................
2.5
Flutter Analysis.......
....
34
................................... 44
3. Experiments
3.1
Test Wing Selection............
ooooooooo
3.2
Static Deflection Tests .......
ooooooooo
..
0
.
3.3
Wind Tunnel Test Setup.........
ooooooooe
.. 0
.
3.4
Natural Frequency Tests*........
3.5
................ 60
.
................ 61
.
.
................ 63
..ooooooooo
0
.
.
.
................ 66
Steady Airload Tests...........
ooooooooo
.. 0
.
.
................ 66
3.6
Flutter Boundary Tests........
ooooooooo
3.7
Flutter Tests..................
.
.
...........
.....
... 67
.. .. ... 67
4. Results
4.1
Static Deflection..........................................70
4.2
Steady Airload Deflection...................................73
4.3
Divergence Velocities.....................................76
4.4
Natural Frequencies........................................78
4.5
Flutter Velocities.......................................79
5. Conclusions and Recommendations.....................................86
6. References and Bibliography.......
.....................
88
Appendices
A. Mode Shape Integrals and Their Numerical Values..................... 90
B. Static Deflection Investigation Results.............................97
C. Steady Airload Investigation Results...............................105
D. Flutter Investigation Results.........................
.............
119
About the Author ............................................................................ 158
LIST OF FIGURES
Page
Figure
2.1
Wing coordinate system............................................12
2.2
Graph of knT vs.
2.3
Flexible wing twist feedback system diagram.......................25
2.4
Lift and drag curves for a flat plate...............................26
2.5
Components of force for a wing at angle of attack.................27
2.6
Approximation to the C
2.7
Approximation for the C curve ....................................
p
2.8
Center of pressure movement with changes in angle of attack.......30
2.9
Spanwise lift distribution for varius twist conditions for
22
................................................
curve.....................................28
29
a straight wing...................................32
2.10
Changes in force due to drag with changes in angle of attack.....33
2.11
Lift distribution in the spanwise x direction using the
Weissinger L-Method..............................................39
2.12
40
Swept forward wing coordinate system..........................
3.1
Static deflection test appartus...................................61
3.2
Static test stand with a moment being applied...................62
3.3
M.I.T. wind tunnel setup..........................................63
3.4
Looking down on a test wing by using a mirror......................64
3.5
Equipment used to illuminate and record test data.................65
3.6
Double exposure showing extreems of flutterr motion...............69
4.1
Static deflection test results for [±15/0]s layup wing............71
4.2
Initial camber (exagerated) in the [±15/0]s layup wing............72
4.3
Steady airload analysis on the four wings with a 15 degree
ply fiber angle...................................................73
4.4
Flutter analysis diagrams for the four wings with a 15 degree
ply fiber angle...................................
4.5
...............
82
Effects of ply fiber angle on flutter and divergence speeds ....... 84
B.1 - B.7
Static deflection theoretical analysis and experimental
data.................. .................
C.1 - C.13
..................... 98
Steady airload theoretical analysis and
experimental data.........................................106
D.1 - D.8
D.9 - D.34
Flutter boundary curves.....................................124
V-g method flutter analysis diagrams......................132
LIST OF TABLES
Page
Table
2.1
Assumed modes used in the Rayleigh-Ritz analysis..................19
3.1
Different laminate layups used for the test wings .................
4.1
Wing divergence speeds............................................77
4.2
Wing natural frequencies..........................................78
D.1
Flutter data, unswept wing.......................................120
D.2
Flutter data, swept wing..........................................122
60
LIST OF SYMBOLS
Symbol
A
stress-strain modulus matrix; aerodynamic matrix
B
stress-bending modulus matrix
b
semi-chord of the wing
C
compliance matrix
c
chord of the wing
Cd
Cf
coefficient of drag
coefficient of force
C
coefficient of lift
Cm
coefficient of moment about the quarter chord
Cp
coefficient of pressure
cp
center of pressure
D
moment-bending modulus matrix
EL
logitudinal Young's modulus
ET
transverse Young's modulus
GLT
shear modulus
g
structural damping
h
vertical deflection
K
stiffness matrix
kiT
non-dimentional torsional stiffness
L
lift
P
length of wing from root to tip
LA
lift associated with bending displacement
LB
lift associated with torsional displacement
LC
lift associated with chordwise displacement
M
moment per unit length; mass matrix
m
mass per unit area
MA
moment associated with bending displacement
MB
MC
moment associated with torsional displacement
moment associated with chordwise displacement
N
force per unit length; chordwise force
NA
NB
chordwise body force associated with bending displacement
NC
chordwise body force associated with chordwise displacement
P
distributed force per unit length
p
distributed force per unt area
Qi
q
generalized enternal force
qi
R
generalized coordinate
S
matrix used to transform K
T
kenetic energy
V
potential (strain) energy; velocity
V,
W
free stream velocity
weighting matrix
w
deflection in the z direction
chordwise body force associated with torsional displacement
dynamic pressure
matrix used to transform K
x
y
cartesian coordinates
z
x
spanwise axis
y
chordwise axis
a
angle of attack
ai
beam bending mode constant
0
warping stiffness parameter
Yxy °
shear strain tensor at midplane
Yi
Rayleigh-Ritz mode
6ij
EXo
delta function
SYo
Ei
strain tensor in the y direction at midplane
beam bending mode constant
strain tensor in the x direction at midplane
9
ply fiber angle; twist angle
Kx
Ky
KXy
A
LT
(
curvature strain tensors
wing sweep angle
Poisson's ratio
chordwise deflection
air density
Rayleigh-Ritz mode shape part that is a function of x
Rayleigh-Ritz mode shape part that is a function of y
CHAPTER 1.
INTRODUCTION
of composite materials in aircraft structures has added
The use
Useful not only for
another design dimension to the aircraft designer.
their
high
strength
vary
ability
to
scheme,
they
the
have
In
attractive.
to
weight
but
certain
particular,
by
previously
forward
giving
by
behavior
deflection
force
made
ratio
the
designer
layup
the
varying
impractical design
options
gained
renewed
wings
swept
the
have
interest because their major drawback, low wing divergence speeds, can be
by
tailored
using
material
composite
in
wing
significantly
improved
construction.
A good discussion of this is contained in reference 1.
This project will draw on the work of four previous experimenters
at M.I.T.;
5).
Jensen,
Hollowell,
Selby and Dugundji
(references
2,3,4 and
Those men worked with some of the same wings that were used in
project.
They made
calculations
and ran experiments
this
the
to determine
wing stiffness and bending-torsion stiffness coupling, the steady airload
deflection behavior and the flutter and divergence speeds at low and high
angles of attack.
range
by
Using their work as a foundation,
investigating
investigating
several
aeroelastic
new
ply
at
properties
its
we will extend
angle
layup
30
degrees
patterns
and
forward
by
sweep.
Primary interest is in the investigation of divergence and flutter speeds
and the wing shapes during those conditions at both low and high angle of
attack.
Low angle of attack
flutter is
investigated both theoretically
flutter
and experimentally while high angle of attack
experimental
airload
only.
deflection
We
also
analysis
extend
by
the work
including
a
of
investigation
Hollowell
realistic
on
non-linear
curve, drag and approximations to three dimentional aerodynamics.
is
steady
lift
CHAPTER 2. THEORY
2.1 Rayleigh-Ritz Analysis
In this research both static and dynamic analysis are done using
We therefore need to formulate an
the Rayleigh-Ritz analysis technique.
equation of motion for the dynamic analysis and then by setting the time
derivatives
equal to zero,
analysis.
Because
of
we can use
the
the same
equation for the static
complicated nature of anisotropic material,
exact analysis is difficult and often impossible.
Therefore I chose an
approximate
mode
method
Rayleigh-Ritz method.
of
This is
and Selby (references 2,
and in
analysis,
the
assumed
or
so
the same method used by Hollowell,
3 and 4)
who were mentioned in
called
Jensen
the introduction
fact, due to the amount of work already done on these wings with
this method I decided to use the same assumed modes im my analysis.
analysis
The
is
perpendicular to the wing in
linear
and
assumes
all
the z direction as shown in
deflections
figure 2.1.
G = PLY FIBER ANGLE
V.
Figure 2.1 Wing coordinate system.
are
Note that the ply fiber angle is
measured
in
the opposite
direction as
compared to the standard composite material direction.
The basic equation.for the assumed modes is:
n
Yi (x,y) q (t)
w(x,y,t) =
(2.1)
i=1
Where Yi(x,y) are the modes.
In our case we have five modes so n=5.
To get the equation of motion we start with Lagrange's equation.
d
dt
ST
K 4 r
av
aT
qr
3q
-
(2.2)
at
Where Q represents the applied loads.
Now we need expressions
for the kinetic
and
potential
energies.
For the kinetic energy of a plate we have:
T =
ff
m (w2
dA
(2.3)
Where m = mass/unit area.
Using equation (2.1) for w we get:
i
S=f f m I iAi
TJ
j
jyj dA
(2.4)
Moving summations and grouping terms:
=
T
i j
(2.6)
=f f Y.my.dA
M.
where
(2.5)
Mqq
The variational potential energy for a plate is:
6v = ff
( N
Se
x
+ N
x
oo
+ M
xy
6
yo
+ M 6K + M 6K
y
y
x
x
Y
+ N
xy
xy
(2.7)
6K xy ) dx dy
where using conventional displacement notation:
*
au
ax
x
= av
ay
Yo
K
y
xo
2
a2w
ax 2
2
au
XY
ay +
aw
2
- 2(2.8)
9v
x
xy
axay
To apply this to anisotropic plates we start with the modulus equations
for the general laminate.
{]
M
B
D
i
K
(2.9)
N = Force/length vector
M = Moment/length vector
= strain vector
K = curvature strain vector
A,B,D are the appropiate modulus matrices
Since this
direction.
is for a plate we assume no strain or shear in the z
For a symmetric
strain, only bending so
direction so
Mx
Y
yxy
laminate,
B = 0.
moments about x any y do not cause
In our analysis all loads are in the z
N = 0, leaving us with, in expanded form:
D11
D 12
D 16
Kx
D2
D2 2
D 26
Ky
1
(2.10)
D6
1
D6 6
xy
Using equation (2.10) for M, integrating with respect to the variational
terms and using equations (2.8) for the curvatures we get:
D1 1
D 12
D 16
S-W,
V = 1/2
ff [-w,xx -w,yy -2w,xy
D2 1
D2 2
D26
XX
yy
(2.11)
D6 1
D6 2
D6 6
-2w,xy
where:
2
w,
=
xx
w
, etc.
2
equation and using
expanding
this
D21Dx
expanding this equation and using
+ 4D
16
w,
xx
w,
We now substitute in
xy
and D2 6=D
62
2
x
+ 4D w,
+ 2 D 1 2 w, x1xyy
,
+ D 22yy
2 2 w, y
66 xy
f f{ D 11 w,
=1
V
D2 1 =D1 2 , D 16 =D61
+ 4D
26
w,
yy
w,
equation (2.1)
xy
j dA
(2.12)
for w bring sumations and q's out of
the integral to get:
i j
V
=I1
qq
i
2
+ D
22
+ 4D
j
ff
y.,
1 xxY., yy
I
D Y
Y ,
xx j
11
+ 4D
66
Y.,
xx
1 xy y,j xy
+ 2D
Yxx
Y.
12 1'xx j'yy
+ 4D
,
y.,
16 i xx 3 xy
} dA
Y.,
Y.,
26 1 yy ] xy
(2.13)
Finally it is rewritten in the compact form:
=-1
i j
Kij qq
(2.14)
where
= ff
K
D
11
Yi,
Y.,
+ D Yi'
22
xx ] xx
+ D1 2
i'xx'
+ 2D
y,
Y.(
j'+
Y.,
1 xx j xy
16
yy
y.,
yy
+4D
y,
,
66 i xy j xy
xxj'yy
+ y,.
Y.,
j xxI xy
+ 2D26 ( Yi'yyj xy
)
(2.15)
Yyyixy
symmetric.
Note that Kij is
Now we can put our expressions for
T and V
in equation
(2.2).
First
for kinetic energy:
Tq r
M..
13
Mij
2
i.
q
qr
a4
q'-
using
qi =
;q
aqr
M
3T
aq
is
+
arq.
r
r
i )
j
and expanding
ir
1 i
;T
-r = - IM
q. + 2
rj i
r 2
since
qi
Mir qi
ir i
(2.17)
symmetric we can sum the two parts:
= IMr
rj
q.
(2.18)
]
similarly for potential energy:
S
Krjq.
(2.19)
finally
3T
d3T
q=
dt
Mrj qj
r
We note that:
aT
these into Lagranges
put
and
equation, using0
and put these into Lagranges equation, using
M
1]
q. +
K.,. q. = Q.
J
13
J
1
r=i:
( j = 1,2,...N)
(2.21)
in matrix form:
M q+
rre
Kq =Q
me
(2.22)
r
This equation will be used to derive all the displacements and motion of
the wings.
The five mode shapes used are
listed on table
the variables have been separated in the form:
Yi(x,y) = 'i(x)
*i(Y)
(2.1)
where
Mode
i.
i (x)
cosh ( -
-
- a
cos ( -EX
- )
{ sinh
Ex
cosh (
2.
-)
)
Ex
- sin
3.
sin (-)
4.
3ix
sin ( --
)
Ex
-cos ( -
Sx
- a2
( -
sinh
sin (
(2
ex
22-
4y
5x
A
A
c
1
2
3
where:
El
= 1.8785104
al = 0.734096
a2 = 1.018466
C2 = 4.694091
Table 2.1 Assumed modes used in the Rayleigh-Ritz analysis.
The first two are cantilever beam bending modes the second two are
beam torsion modes and the fifth is
notice
that
modes
3 & 4 do
cantilevered
plate
at the
You may
a chordwise bending mode.
not meet the
root where
w,
x
boundary
is
zero.
conditions
But the
for a
error
small away from the root and an aspect ratio correction for this is
is
made
3 goes
Jensen in reference
terms.
matrix
in the stiffness
through in
detail the algebra for working out the mass and stiffness matrix terms
In this
correction factor.
torsion stiffness
and he also derives the
report only the results are stated.
Mass Coefficients
M
22
= mc£I
mct
I
12 3
M55
44
mc£
12
12
M..
13 =0
M
2
4mc£
I
45 I5
33
M
=mctI
M
4
i
j
Stiffness Coefficients
D
D
c
11
K
=-I
11
£3
K
I6
22
=
2D
26
K
=-I
2
23
K 12 =0
16
22
2D
K
14
=-
K3 4 = 0
10
4D 16
K
11
-
35
32
4D
16
2.2
24
7
K
25
=-
66
c2
33
4D
K
55
=
c
11
452. 3
22
c
3
20
64D66
I
5
+
3ck
c
2T
37/2
2
16D 2 6
iw/2
64D 22
+
19
2
1T
26
I
17
2
+
k
66
32
32.
45
k
15
16
2
4D 16
16
I
13
ci
4D
12
ct I9
16D
I
c£
44
12
8D
16
I
8
2
8D
15
I
2D
2D
13
c
11
3
I
23
2
21
Where the original values for K33 and K44 were changed using the aspect
ratio
correction
worked
out
by
and
Crawley
Dugundji
(contained
in
reference 4).
where:
D11c
8 =
2
48D 66
2
8 in figure 2.2.
And the values of knT ( n = 1,2 ) are plotted vs.
The integrals and their values are listed in table 2.1.
using
the
The engineering constants are for Hercules ASI/3501-6 graphite/epoxy
for
For
our
particular
wings,
values
of
Dij
calculated
were
material constants listed below.
EL = 98 x 109 N/m2
2
ET = 7.9 x 109 N/m
VLT = 0.28
2
GLT = 5.6 x 109 N/m
wing thickness = .804 mm
density of wing = 1520 kg/m 3
loading
out-of-plane
differences
loading).
in
(see
reference
the engineering
constants
5
for
for in
an
explanation
on
the
plane and out of plane
2.05
2.00
1.95
1.90
1.85
1.80
1.75
1.70
1.65
1.60
1.55
0
E-3
5
0
E-3
5
10
15
20
25
30
35
40
45
50
E-3
30
35
40
45
50
E-3
BETA
7.5
7.0
65
S6.0
5.5
5.0
4.5
10
15
20
25
BETA
Figure 2.2 Graph of knT vs.8.
2.2 Static Deflection Analysis
For static loading, all inertia terms in equation (2.22) are zero
so we have left:
(2.23)
Kq = Q
Q is the modal force where:
Qr =ff py rdA
Qr
(2.24)
r dA
where p is a distributed load per unit area.
For our tests the load was applied at
c/2
Qr =
-c/2
so we have:
£
I
f
x = £
p(x,y)
6
(x-£) Yr(x,y) dA
0
c/2
= f P(£,y) Yr(£,y) dy
(2.25)
-c/2
where P is the load per unit length.
For the bending
load
the apparatus used was assumed to apply a
the y direction so we can take P out of the integral.
load constant in
c/2
Q = P
r
f
Y (a,y) dy
r
(2.26)
-c/2
Now we
need only
insert the
five modes
to get the five values
of Qr.
They are listed below.
Q1
= 2Pc
Q2
=
-2Pc
Q3 = Q4 = Q5 = 0
For
the
torsional
load
the
test apparatus
was
assumed
to apply
load
linear to y such that:
p = ay
where a is a constant.
Our equation becomes:
c/2
Q = af y y (,y)
r
r-c/2
-c/2
dy
(2.27)
Again,
putting in
the five modes we get:
Q1 = Q2 = Q5 = 0
ac
12
Q3
2
2
ac
12
4
The values for Q were calculated
solved
for
the
vector q.
The values
to get the analytical
(2.1)
equation
column
(2.23)
and put into equation
of q were
which was
then put into
The results are shown
deflection.
together with the experimental data on figure B.1 to B7.
2.3 Steady Airload Analysis.
When
put
in
airforces,
generates
an
airstream
indeed this
at
is
a
given
angle
its purpose.
of
attack
a wing
These airforces
not
only support the weight of the airplane but they also bend and twist the
wing itself.
This deformation of the wing,
angle of attack,
in
return,
changes the airforces on the wing.
by changing
The result,
shown in
figure (2.3) is a simple feedback system.
a
0
c fopressure
curve
Figure 2.3
dynamic
ressure
force(a)
wing
stiffness
stiffness
Flexible wing twist feedback system diagram.
the
When
the
loop
In
deflection.
increased
beyond
airforces
and
divergence.
converges
certain
a
to a
deflection
value
increase
the
loop
without
we
does
have
the
when
theory analysis
linear
critical
certain value
not
limit.
the
final
airspeed
converge
This
is
is
and
called
In the real world there are limits to the actual increase in
deflection and airforces.
come
They
to distruction
about due
of
the
wing, non-linear stiffness of the wing or non-linear increase of airload
with increases in the angle of attack.
non-linearity,
especially
at
With our wings the most important
airspeeds
at
and
below
the
divergence
airspeed, is the non-linear increase of airload with angle of attack.
For the steady airload analysis we put in this non-linearity by
using the lift and drag curves for a flat plate from reference 10 shown
in figure 2.4.
reference 10 data
0.8
C
0.6
CD, C
L
0.4
CD
0.2
0
4
8
12
16
20
angle of attack
Figure 2.4
Lift ands drag curves for a flat plate.
The force that deflects the wing is the force perpendicular to the
wing (in the z direction).
This force has a component from lift and drag
dependent on angle of attack as shown in figure 2.5.
Lift
Resultant
Force
Vs
Drag
Figure 2.5
Components of force for a wing at angle of attack a.
From figure 2.5 we get:
Cf = CY cosa + Cd sin a
For the lift component of force, I made a Cf(lift) vs.
(2.28)
a curve.
This
curve was approximated by a polynomial of the form:
n
Cf(lift)
[
i=1 A
a
(2.29)
For our purposes
gave a sufficiently accurate curve.
n = 4
Both the
coefficient of force curve and the polynomial approximation are shown in
figure 2.6.
4
C
= -37.7072a
f(2.30)
3
+ 42.472a
- 23.167a
2
{a
+ 6.6746a
in radians}
(2.30)
0.6r
Cf(lift)
reference 10 data
approximation
-
0.2
U
I
20
16
12
8
4
0
angle of attack
Figure 2.6
This
gives
the
Cf curve.
Approximation for the
section
coefficient
of
force
from
lift
as
a
function of angle of attack but we also need to know the dependance on x
and y, or in other words, force distribution on the wing. In reference 4
Selby showed the lift distribution of a rigid wing.
guide
for
modeling
the
force
distribution
polynomial approximation was used.
on
our
This.was used as a
wing.
Again
a
The distribution in the chordwise y
direction and the approximation are shown in figure 2.7 as coefficients
of pressure versus chord.
equation
Approximation
in
distribution
force
for
the
chordwise
y
direction:
C
0
= C
3( 0.5
-
y
2
(2.31)
p avg
p
reference 4 data
approximation
0.2
0
0.4
0.6
0.8
1.0
(y + 0.5)/cord
Figure 2.7
Approximation for the Cp curve.
Both the approximation and the theory give 25%
chord as
the center of
pressure in the y direction. Reference 10 shows the coefficient of moment
about the quarter chord vs. angle of attack. From this we can calculate
the center of pressure movement with changing angle of attack by using:
C
=(py cp -y
m
.25c
Cit
where ycp is the center of pressure location.
(2.32)
To approximate
center of
a modification was made
this,
approximation are shown in
pressure movement and its
The
(2.31).
to equation
figure
2.8.
Approximation
for
equation
distribution
force
in
chordwise
the
y
direction with a correction for cp movement:
( 3.5 - 5.71a )*( 0.5 - y
C = C
p avg
p
2.5
(2.33)
+ 1.63a
reference 10 data
0.4
-
-
approximation
8
12
16
20
angle of attack
Figure 2.8
Center of pressure movement with changes in angle of attack.
In the spanwise x direction, the lift distribution was calculated
using lifting line theory.
I used the matrix equation given in reference
7.
rsinro
i
S 81 sin1+F
N
sinro l
c I
(2.34)
a
where:
0
= 2w
c = chord
£ = semi-span
x
m
=
cos
-
( 2n
r =
(
-
m
1
)
n = column number
m = row number
The lift
at
2 degrees
distribution was calculated for three cases, a rigid wing
angle of attack,
a wing with positive
twist of 4 degrees
with the root angle of attack at 2 degrees and a wing with negative twist
of 4 degrees with the root angle of attack at 6 degrees.
These three
cases span a good part of the low angle of attack deformation conditions
for our wings.
The
three cases along with the approximation used are
plotted in figure 2.9.
The
same approximation was
used in
all
cases.
Note that C£a(root) = C/a(root)
Approximation
equation
for
force
distribution
in
the
spanwise
direction:
C
f
= C
f avg
1.111 1 -(
x
R
)
9(2.35)
}
(2.35)
x
= 2 deg. constant
lifting line
- - approximation
(
-
8
C (root)
4
0
0
0.2
0.6
0.4
0.8
1.0
x/length
S
C
= 6 deg. root to 2 deg. tip
lifting line
approximation
(root)
0.2
0.4
0.6
0.8
1.0
0.8
1.0
x/length
C t(root)
0
0.2
0.4
0.6
x/length
Figure 2.9 Spanwise lift distribution for various twist conditions for a
straight wing (A = 0).
for
Finally,
drag I used
the
figure 2.10 with its approximation.
curve
from reference
10 shown in
Drag was assumed to be a function of
angle of attack only and not of x and y .
Approximation equation for force due to drag:
Cd sina = 3.5a 3
d
Cf. (drag)
(2.36)
0.20
reference
10 data
approximation
0.15
Cf (drag)
0.10
0.05
0
8
4
12
16
20
angle of attack
Figure 2.10
Putting
Change in force due to drag with changes in angle of attack
all the approximations
distribution on the wing where:
p=Cp
p
p
.Axy
Z'c
together we get a two dimentional
force
C
4
A
=
+ A 3a
3
(
*1.11[ 1 -
+ A 2a
2
+ A1 a
] { (3.5
+ 1.63a
25
[ 0.5 - (
- 5.71a
}
3
(2.37)
+ 3.5a
For
a,
vast
the
majority
the
of
twist
()
is
the
from
first
assumed
torsion mode so I used equation (2.38) for a = aO + 0 *
q3
) sin(
+ sin- ( -
a(x) = a
'
)
(2.38)
This completes the distributed force coefficient term.
Multiply this by
This
dynamic pressure and we get the distributed force per unit area.
was put in the Rayleigh-Ritz analysis to get the modal force Q.
c/2
X
Qr =f
0
f
(2.40)
f(x,y) y(x,y) dx dy
-c/2
The equation was solved on a digital computer by numerical integration.
Then equation
(2.23) was solved for q.
Solutions were calculated
for
three airspeeds 5, 11.5 and 16 meters per second at angles of attack from
2 to 20 degrees
at 2 degree increments.
If
16 m/s was higher than the
wing divergence speed, I omitted the calculation at that speed.
m/s three iterations were sufficient,
for 11.5 m/s five iterations were
sufficient and for 16 m/s up to seven iterations were used.
for a straight wing (A = 0)
For 5
The results
are plotted along with experimental data in
figures C.1 through C.13.
2.4 Wing Divergence Analysis.
Basically
a
wing
diverges
when
the
aerodynamic
forces
from
increased angle of attack due to twist are stronger than the resisting
forces
from
the
wing's
bending also causes
torsional
stiffness.
When
wings
are
swept,
changes in angle of attack while bending stiffness
resists these changes.
To calculate divergence, therefore we need to relate the stiffness
forces to the aerodynamic forces.
In matrix form the torsional stiffness
equation is :
If
(
= [ c zE
{ 0
[ C
I
M
(2.41)
the forces and moments can be combined to make a force at a specific
chordwise point we get:
S) =
[C
( F )
(2.42)
The aerodynamic forces for a station along the span are:
a
q = dynamic pressure
= cC ,q
)
(2.43)
In matrix form we have:
L } =q
Where
the
[
W
]
is
appropiate
cC
}
(2.44)
a weighting matrix for the stations chosen and contains
amount of spanwise
length to make the running
lift for the area covered by the station.
in equation (2.42) so we have:
lift
.the
In this case lift is the force
So } = [
[
c ]q
]
}
cC
(2.45)
By use of an aerodynamic scheme we will get:
(2.46)
)
a =A(cC
In matrix form:
Where A is the aerodynamic operator.
Sa
=
[ A]
(2.47)
cC
or:
(2.48)
a
C,}=[A]
putting this into equation (2.45) we get:
)
=
Where
a0
q[c[
a
We note that
[
a
{a
(2.49)
+0
is the rigid or root angle of attack.
To calculate divergence,
Sa
-1
[A]
=
{e }
we set
a0
=
0,
so we get:
(2.50)
and
-1
1
[
o }=[ C
-{
q
[
]
-1
(2.51)
{}A
This we recognize as a characteristic
value equation where for the first
characteristic value, say X, we have:
= --
(2.52)
(divergence)
This is the form of the solution to the divergence problem.
only determine the three matricies
[ C ],
[ W ] and [ A ].
We need
Because the
compliance matrix will be made to match the aerodynamic matrix we will
determine the aerodynamic matrix first.
To derive the aerodynamic matrix I used the Weissinger L-Method.
This method and its theoretical
foundation are outlined
in
reference
7.
The final matrix form for lift symmetric about the fuselage is:
a{
=[ A ]
{
}
cCS
(2.53)
DeYoung and Harper is reference 11 write this equation in the form:
(m-1)/2
a
The
avn
v
=
a
n=1
terms
vn
are
G
(2.54)
n
the
form of the chord times
terms
the
lift
of
the
aerodynamic
coefficient.
matrix
and
Gn
is
a
They have graphed values
for
the
aerodynamic matrix terms
versus wing geometry giving a 4 x 4
Although the swept wing was of primary interest in
aerodynamic matrix.
the divergence analysis, I constructed an aerodynamic matrix for both the
straight wing and a 30 degree swept forward wing ( A = 0, -30 ). With the
aerodynamic
matrix
and
known
angle
of
solution of the four equations gives
attack,
a
simple
simultaneous
the section coefficient of
lift.
The results are shown for constant angle of attack for both straight and
sweep forward wings in figure
2.11.
The lifting
line results
for the
straight wing are coplotted for comparison.
The
four
stations
used by Harper
and
DeYoung
are
the
four
so
called Multhopp stations and are at:
x
= .9239, .7071,
for the stations
vortex
.3827, 0.0
1, 2, 3 and 4 respectively.
chord while
at the quarter
This method puts the bound
meeting boundary conditions
at the
This is equivalent to saying the force is at the
three quarter chord.
quarter chord while the angle of attack is measured at the three quarter
chord position.
This gives
us the
necessary information to match the
compliance matrix to the aerodynamic matrix.
Also reference
7 gives the terms
in [E] when using the Multhopp
stations.
To
calculate
the
system in figure 2.12.
compliance
matrix we will use
the
coordinate
x (root)
0.6
0.4
0.2
0.8
x/length
= constant
m(root)
i/length
Figure
2.11
Lift distribution
Weissinqer L-method
in
the
spanwise
x
direction using
the
V,
A
---
y
y
Figure 2.12 Swept forward wing coordinate system.
In our case all
loads
can be
considered point loads applied at
quarter chord in the y-axis and at the Multhopp stations in the x-axis.
This reduces equation (2.24) to:
4
j=1 F.x.,y.
expanding this:
(2.55)
\
1
1(x ,
1(x2
2 ).
Y1( x4'
4)
2(x4
4)
F
Q2
2 x2' 2 "")
75(Yx2 2 ..
5
{
Q} =[R]
Now putting
angle 0.
e
w
5 x4'y 4
{F},
[R],
F
so in
4
short form:
F
(2.57)
that aside for
the moment,
get a
let's
relation
for
twist
Using small twist angle assumptions we can say:
= -
(2.58)
-
= -
For
that premultiplies
call the matrix
Let's
F2
cosA - a
(2.59)
sinA
we substitute in equation (2.1)
5
yi (x,y)
ai (x,y)
a(
0=-
- cosA +
sinA ) qi
(2.60)
i-=1
To get the twist at the four Multhopp stations we use the x-axis position
of the station for
3j
and three quarter chord for
index for the Multhopp station.
This gives us:
j where j
is the
e
5
j
= -
ay
yi (x.,y
++cosA
x.,y
sinA)q.
-
(2.61)
i=
i=1
in expanded form
a1(Xl,
ay2(x1
)
I
01
ay 5 (x 2 ,y 2 )
ay2 (x2 f
)
ay1x 1
x2
ay
2)
ay
2
1xy 4)
a2(x4
4
a5(x4,y
....
cosA
4)
ay
e4
a 2(x IYl
....
a5s(x11Y)
ql
ax
a
ay 2 (x
2 ,'
2)
....
ax
sinA
a7 1a.
(x2'Y
2)
ay 1 (x 4 'y
4)
ay 2 (x4'Y 4 )
a3E
Calling the matrix
....
Dy5(x4
93F
37'
that premultiplies
{q}
in
4)
equation
92
q5
(2.62),
[S],
we
have in short form:
S)
=[ s] {q}
(2.63)
Now using equation
(2.23) where we premultiply by the inverse of [K].
q}= [K
(2.64)
Q}
SK
putting this in equation (2.63) we have:
{
}= [s]]
K
{Q
(2.65)
putting equation (2.57) into equation (2.65)
{
}=[s][K]
[R ]{F
(2.66)
We can now see that our compliance matrix is of the form:
[c] =[s][K]
[R]
(2.67)
Finally, we put equation (2.67) in equation (2.51) giving us:
1
){
-1
[s][K]
-1
[R][W][A]
{}
(2.68)
q
This
equation
characteristic
was
solved
in
a digital
computer
value gives the divergence velocity.
and
the
first
With some of the
wings with negative bending torsion coupling, for the straight wing case,
the divergence velocity was imaginary.
This indicates that according to
linear theory .the wing will not diverge. The results of the analysis are
shown together with experimental data in table 4.1.
2.5 Flutter Analysis.
For flutter analysis,
this method,
structural
V - g
I used the well known
damping
(g)
is
introduced into
method.
In
the equation of
motion. since solutions of the equation of motion represent the neutral
When
g
solution,
condition
stability
is
positive
we
see
when
that
Flutter occurs when g is
stability.
g is
the
negative
damping
is
is
wing
required
for
stable.
neutral
equal to the actual damping of the
structure.
Assuming
motion
harmonic
q(t)
= q
eiLt
),
the
equation
of
motion is:
2
-
SK
ist
itQ
)
M
(2.69)
First we need to derive the unsteady aerodynamic forces in terms
of Q in equation
work (W)
(2.22).
We will do this by deriving the variational
and put that in terms of Q by using equation (2.70).
5
I
Q q
n-1
r
6W =
(2.70)
r
The general form for 6W is:
1
6w = f
0
The
term
respective
c/2
(2.71)
f p 6w dydx
-c/2
p6w
can
be
displacements;
separated
lift,
into
moment
three
and
components
camber
force
and
about
their
the
midchord (L6h, M60 and N6).
where:
c/2
L =
p dy
f
-c/2
c/2
yp dy
M = - f
-c/2
c/2
N =
I
-c/2
5 p dy
(2.72)
~PI
V
+1
4-91?'
Figure
2.13
convention.
Swept wing
coordinate
system and
the
displacement
sign
Using
figure
figure
2.13
2.12
for
(reproduced here)
the
h
(x) q1 +
=
1
=-
55(x)
sign
displacement
assumed modes to the different
2 (x)
3(x)3
for
the
swept
convention,
wing geometry
and
then
the
displacements we can write:
q2
+
relating
(2.73)
4(
)
q4
dh
cosA + d sinA
}
(2.74)
(2.75)
q5
Using equation (2.73) we can write equation (2.74) as:
8e
-
{
[ 03(x) q3 +
4 (x)
q4
+ - q1 + -
cosA
q2
sinA
}
(2.76)
Putting equation (2.72) in equation (2.71) we get:
W = f L Sh dx + f M 6e dx + f N 6~ dx
0
0
Now putting equation (2.73),
rearranging terms we get:
(2.77)
0
(2,75), and (2.76) into equation (2.77) and
d1(x)
6W =
{
fL
1
(X) dx
sinA
-
fM
d
f
1
dx
6qi
d
1
0
+
f L
2
-
(x) dx
sd#
2 (x)
sinAI M
d
dx
}
6
q2
0
cosA
f M43(x) dx
6q3
0
f M
C
0
f
+
0
4
(x) dx
Sq4
(2.78)
N t5 (x) dx 6q
5
Now, taking the relations worked out by Spielberg for 2-D incompressible
aerodynamic theory
in
12, and adapting them to our swept wing
reference
case, we get equations for lift, moment and camber force.
B
Ab
M = wp2be
N =
Where p is
wrp
2 3 it
be
( M h + MB
(N
h
Ab
+ N
B
the air density, a is
Cb
+ MC
) cosA
+ N
) cosA
Cb
the
(2.79)
oscillating frequency, b is
the
functions
the
and
semichord
LA,
LB ,
LC ,
non-dimentional complex functions of reduced frequency
are
etc
MA ,
k = Wb/V, given
by:
L
A
L
L
i
+
k
B
C
2C (k)
-i
=1
k
2C(k)
SC(k)
X
+
k
k2
i
1
12
=
-i
2C(k)
2
C(k)
3k
k
C(k)
k
M
M
B
NA
N
1
i
8
2k
C
N
N
=---
B
C
+i
C(k)
C(k)
2
2k +
k
i
-1
2k
C(k)
+
6k
1
12
C(k)
3k
i
3k
C(k)
6k
=----
=-+i
1
=--i
36
j C(k)
+
1 8k
k
C(k)
1
2
k
C(k)
2
3k
+
1
2k
2
C(k)
2
3k
(2.80)
where C(k) is the Theodorsen function.
49
These equations assume harmonic motion such that:
h(x,y,t)
h(x,y) e
=
iwt
(2.79),
Putting equations
, etc.
(2.73),
(2.76)
(2.81)
and
(2.75)
in
(2.78),
equation
using equation (2.70) and rearranging terms we get for Qi:
Q =
pw 2b
I [ cosA
3
iwt
LA
- f
*
2
(
)
(X)
(x)
d
d
- cosA sinA M
dx
A
dx
dx (x)
1(x)
d
- cosa sinA L B
f 1 ((X)
0
(
)
i
q1
0
#2(x) dx - cosA sinA LB f
1
(x)
d#2 (x)
d
dx
0
da 1 (x)
d 1(
- cosA sinA M
2
dO (x)
2
l (x) dx + b cosA sin A M
L
+ [ cosA
dx
2(x) dx
0
Sd
2
+ b cosA sin A M
B
0
(x)
d2 (x)
d
dx
dx ] q 22
cos 2 A
c
LB f
2
cos ALB
c
L
+ [ cosA
=
Q2
I
L
b
-
(x)
(x)
1 x)
3
)
f
(x)
4
1
(x) dx -
dx
2
-bcos
4
B
dx
d1
A sinA
5
(x)
(x)
B
c
f(x)
d
4
dx
(x)
dx - cosA sinA Mc I
d)
C
dx
05
dx
]
q
5
2 3 iwt
wpw b e
d
LA
{
(x) dx ] q 3
3
1
d
(x)
(x)
d
b cos A sinA
c
cosA - f
2(x)
- cosA sinA MA
W1(X)
dx - cosA sinA LB
d2(x)
d02 Wd
dx
/
I
2 (x)
(x)
dx
dx
1(x) dx
0
d
+ b cosA sin 2 A M
2
L
+
[
cosA
b
-
2 (x)
2
- cosA sinA MA f
B
d2
dx - cosA sinA LB /
LB J
dx
2x
2
2
(x) d
0 (x) dx + bcosA sin 2 A %
(x)
x ] q
1
d rx
d
d
d(x)
dx
(x)
dx
dx
1
dx
( d
2
(dx)
] q2
2
2
(x)
x)
A L
LBB
cos A L
(x)
dx
2(b
b cos A sinA
cos
A sinAM
0
LC f
f
d(x)
2)
f
3(x) dx
.q
2
0
+[ cosA
+
MB
2(x)
5(X)
d
W
do 22 (x)
dx - cosA sinA MC f
di
5 (x) dx ] q5
2 3 iwt
- wPW
Q3
b e
2
cA s
E A
S[
3 x)
(x) dx1 x)x
2
b cos A sinA
£
dl (x)
MB f
4)3(X)
b cos A sinA
MBB f
c
0
X
c1
(x)
d
+
cos A M
c
3 (x)
A
2(x)
2 dx-
0
_ [ b cosAA
-
2
M f 03(x)
0
dx ] q3
£
-
b
MB
c
f
0
3
(x)
4 (x ) dx ] q 4
3
)x)
dx ] q1
d
2
d2
)
dx ] q2
+
cos
A
c
M f
3
(x)
(x) dx
]
q5
0
Q4= p2 3 iwt
*
Q 4 =-- Wpbe
cos 2A
1BJ
(x
A MA f4
2
a
(x)
dx
d
(
4
]
0
2,
2
+ cos A
+
MA f
0
(x) dx
2
t4(X)
b cos A sinA
c
MB f
0
4
(x)
_ 2(x)
d
dx ] q2
3
b cos A
-2
B
f
*4()
3
b cos A
--2
MB
f
(
03(x) dx ] q 3
c
2
)
dx ] q4
c
cos 2 A
S
MC f
0
+
4
()
5 (x)dx
q
5
2 3 iwt
5
= WpW b e
S cosA
d
NA
5 x)
1()
I
dx - cosA sin
N
5
x
x
r
d
x
i
dx]q
Sd2~(x)
+[
sA
f
A
5
0
-
[oL
2
NB
(x) 4 2 (x) dx - cosA sinA NB f
0
5
3 (x)
(x)
5)
d
dx
]
q2
dx] q 3
(2.82)
0
2
[ cos A
B
5 x) *4(x) dx
] q4
0
+
Nc f
5 (x)
dx] q5
0
to
Inorder
with
integrate
respect
to
the
variable
we
x
use
the
substitution;
(2.83)
x = E cosA
This makes the limits of integration: 0 to X/cosA or 0 to
original distance in the unswept case.
1, which is the
Now we write Q in the form:
5
q
=
wp2b3e
i
t
I Aij q
j=1
(2.84)
where the terms of [A] are:
2
cos A
b
LA
b
A
A11
11
I
2
- cos A sinA L
1
B
I
2
- cos A sinA M I
A 24
24
2
2
1
I
+ b cos A sin A M
Bt 25
A
A
=
=
12
- cos 2AsinA L I
- cos 2 AsinA M I
+
B 26
A 27
13
3
cos A
c
14
3
cosA
c
LB
+
2 -9
B
A22
22
2
- cos A sinA MC 134
34
33
I
I
27
-cos
2
2
sinAM I
+ b cos A sin A
A 26
MB
X
2
2
- cos A sinA L
2
3
A
3
24
=
=
cos A
--
5
LLB
B
cos A
-A
L
c
B
28
30
B
I
35
- cos
1
2
2
+ b cos A sin2A M - I
B
36
A2
MB
32
31
A2 1 = - cos 2A sinA L
21B
2
cos A
b
LA
b
A
a
3
b cos A sinA
2
cos A
b
C
15
3
b cos A sinA
M
c
B
2
2
b cos A sin A
I37
37
I
3
b cos A sinA
MB 138
3
+ b cos A sinA M
39
c
B 140
2
sinA M I
A 35
28
I
-I
N VUTS V SOO
N
LSv
q
V SOD
0
-
=
sv0
V soo
so
V
q
0
I
W
VUT
s
+
6£
V
LE
w
0
V£S o o
V
=
0
V
W VI
V Soo q
V IOW
=
V soo
0
=
VT
V soo q
=
V~
0
EI '6W
8EI HN ~
o
S VuTs V soo q
£
LEI
0£ I
6Z
gW V u T s
o
+
V soo q
IVI3
vu
'6vW
V£ Soo
0f
V soo
Z V
LE
VSo£
ZI DN VUTS V ,soo
LI
DDq
v
q
s O
r
_ Sq
A5 2 =
52
2
cos A
N
b
A
b
I
2
- cos A sinA N I24
B 42
41
53
cos A
c
B
43
54
cos 3 A
N
B
c
14I
44
A
A5 5
55
2
cos A
bos NC
C
b
I1
(2.85)
5
The integrals and their values are listed in table A.1.
From this point on, the flutter analysis for the swept wing is the
same as the analysis for the straight wing as give by Hollowell and
Dugundji in reference 5.
Equation
is
(2.84)
put in
(2.69)
equation
and both sides are devided by
eiwt.
K q - w
In
the
damping
normal
2
2 3
M q = wpw b A q
fashion
for
the
(2.86)
V
- g
(g) is introduced by multipling
flutter analysis,
K by
(1 + ig).
structural
We define a
complex eigenvalue Z as:
1 + ig
2
(2.87)
2.7
we define a new matrix B as:
B = M + rpb
3
A
(2.88)
Finally putting equations (2.87) and (2.88) in equation (2.86) we have, a
standard form, complex eigenvalue problem:
{
-Z K I q = 0
(2.89)
This equation was solved on a digital computer.
of
The
Selected values
reduced frequency k were used to calculate the aerodynamic matrix.
eigenvalues
Z were
used
structural damping required g,
to get
the
frequency
oscillating
w,
the
and the velocity V, according to equations
(2.90).
W
1
Re(Z)
g -
Im(Z)
(Z)
Re(Z)
(2.90)
The flutter velocity was chosen as the velocity at which the structural
damping required to maintain neutral stability became zero on any of the
five modes.
The associated
simplify calculation of
value of w is
the flutter frequency.
the Theodorsen function C(k),
I used the R. T.
Jones approximation given in equation (2.91).
C
2
0.5 p + 0.2808 p + 0.01365
p + 0.3455 p + 0.01365
( p = ik
To
}
(2.91)
Note that by using large values of k the output frequency will be
the natural vibration frequency of the wing in an atmosphere of density
p.
This technique was used to get the theoretical natural frequencies.
This analysis was done for all thirteen wings.
plotted in figure D.9 through D.34.
The results
are
CHAPTER 3. EXPERIMENTS
3.1 Test Wing Selection
The criteria defining desireable characteristics of the test wings
are the same criteria used by Hollowell in reference 2.
they are:
a) The wings should have a wide range of bending-torsion stiffness
coupling.
b) The wings should have constant chord, thickness, sweep and zero
camber.
c)The wings
should flutter or diverge within the
0-30 m/s speed
range of the available wind tunnel.
d)
The wings should be small enough to be made with the available
equipment for manufacturing graphite/epoxy plates at M.I.T..
e) The wings should not sustain any damage under repeated large
static and dynamic deflections.
To
bending-torsion
give
a
coupling
good
and
cross
stiffness,
section
both
of
balanced
laminates were made at three different ply angles.
the
range
of
and
unbalanced
In table 3.1 we can
see the range covered.
[ 02/90] s
[+15 2 /0]s
[±15/0]s
[T15/0]s
[-152/0]
[+302/0]s
[±30/0]s
[T30/0]s
[-302 /0
[+452/01s
[±45/0]s
[T45/0]s
(-45 2 /01s
increased
negative
bending-torsion
Table 3.1
coupling
s
increased
positive
bending-torsion
coupling
Different laminate layups used for the test wings.
--i.--
- .
~-------*--X--~X-l --..
~-3~-C--C_-
The
[02/90]s
- -~----
serves
as
I-
the
only
uncoupled
bending-torsion coupling means when the wing is
direction
the
wing will
twist
is
in
the
the
same
positive
twist
direction
as
twist direction
attack.
Because of the layup convention shown in
on
outside
plies
result
in
negative
example.
bent in
positive
the
---
Positive
the positive
direction.
positive
z
The
angle
of
figure 2.1,
+0 fibers
bending-torsion
stiffness
coupling and vice versa.
3.2 Static Deflection Tests
The goal for the static deflection tests was
wings under static loads of pure force and pure moment.
to test the
Also the loads
should be large enough to cause large deformations, inorder to identify
any non-linearities in
the stress-strain relations and thus identify the
limitations of our linear analysis.
The
static
deflection
Figure 3.1
setup
is
shown
in
fig
3.1.
Static deflection test apparatus.
It
is
I
constructed
of
sturdy
wood
with
beams
insides of the top horizontal beams.
horizontal
deflection
vertical deflection.
while
a
metal
rulers
to
attached
the
These rulers were used to measure
carpenters
square
was
used
to measure
To minimize measuring error we extended pins from a
balsa wood clamp at the wing tip.
threads ran from
two eyelets
on the
balsa wood clamp around pulleys clamped to the side of the test apparatus
Figure 3.2
Static test stand with a moment being applied.
and down to weight holders.
One eyelet was at the wing leading edge
while the other was at the trailing edge.
and weights were put on
the same
side of
To apply a force the pulleys
the
test apparatus
while
to
apply a moment the pulleys and weights for the leading edge were put on
one side and those for the trailing edge were put on the other side. (see
figure 3.2)
We applied force in
then in
cm.
double increments
increments of approximately 0.2 N up to 1.0 N
till
a bending deflection of approximately
12
Moment was applied in a similar manner with initial increments of
.0075 NM till approximately 0.2 NM.
This caused a twist of from 8 to 22
degrees depending upon the torsional stiffness of the wing.
3.3 Wind Tunnel Test Setup.
We did the wind tunnel tests in
This
the M.I.T.
tunnel has continuous flow with a 1.5 by
Figure 3.2
acoustic wind tunnel.
2.3 meter
free-jet test
M.I.T. wind tunnel test setup.
section 2.3 meter long, located inside a large anechoic chamber.
tunnel speed range is
continuously variable up to approximately
Velocity was sensed by a pitot tube in
The
30 m/s.
the throat immediately before the
test section and registered on a electrical baratron.
The test setup is
shown in
figure 3.2.
The wing mount consists of
a turntable machined from alluminum and mounted on a rigid pedestal.
mount for
the
wing
is
attached
to
the
turntable allows rotation of the wing to
top
of
the
turntable.
of 15 degrees from -45 to +45 degrees of sweep.
sweep
positions
mounted
increments
The photo shows the wing
In our testing only the -30 and 0 degrees of
were used.
Figure 3.4
Slightly
below
the
base
of
the wing we
Looking down on a test wing by using a mirror.
a flat disk
to provide
smooth airflow
background for the vertical photos,
past the model,
rod and also a place to mount a terminal
wires.
The disk was also labeled for each test to identify the still
pictures.
A
typical
photo
a good
a place to mount the angle of attack
control
video
The
angles of attack from -4 to 20
degrees while the top of the wing mount allows wing sweep in
in -30 degrees of sweep.
The
taken
during
for the strain gauge
a
test
is
shown
and
in
;-- -- - -rr---- -- ~--r
,-----~--
~--'i -- I;
We hung a mirror over the wing to get the pictures looking
figure 3.4.
down on the wing.
still camera,
down" the
-----r~rr -
strobe
motion
were
floodlights
the location of the
Figure 3.5 shows
and floodlight.
during a
used.
flutter
By
checking
necessary to
When it was
test we
the
strobe,
used a
strobe
video camera,
frequency
"slow
otherwise
we
could
determine the wing flutter frequency.
The scale on the disk was used to measure wing tip bending and
twisting.
It is graduated in 1 cm increments.
Since viewing angle and
position affect the picture you see when looking at the test wing through
the
mirror,
tests
and calculations
were made
displacement to the actual displacement.
These
to
adjust the apparent
adjustments
were
then
applied to the data readings off the pictures.
Fig.3.5
Equipment used to illuminate and record test wing movement.
- - I--------
-
3.4 Natural Frequency Tests.
Hollowell,
Jensen,
and
Selby
(references
2,3
and
theoretical analysis using the
gives
results
accurate
torsional
natural
test.
natural
frequency
the
for
tests
we
and
Therefore
frequencies.
simple
did a
wing mounted
With the
first
five Rayleigh-Ritz
modes
and
first
second
instead
initial
the wind
for
already
Also they have
tested many of these wings for their natural frequencies.
shown that the
4),
tunnel
bending
of
doing
extensive
deflection
vibration
test we gave
it
an
initial deflection in twist and bending, released the wing and recorded
the oscillations on the strip chart recorder.
The resulting oscillations
contained strong first bending with weak second bending and first torsion
oscillations.
3.5 Steady Airload Test.
The steady airload tests were to be run at zero sweep and wind
However after investigating the results and
speeds of 5, 10 and 15 m/s.
repeating selected tests we determined that the baratron on the original
than the actual
tests was indicating a lower airspeed
Using
a different
tests
we determined
tests were 5,
and accurately
that
the
tunnel airspeed.
calibrated baratron during the
actual
tunnel
airspeeds
on
the
repeat
original
11.5 and 16 m/s within a tolerance of 0.5 m/s.
To run the test,
first
we set the
then using
tunnel speed,
the
strain gauge readings we zeroed the angle of attack by setting it so that
the average bending and torsion gauge readings were zero.
(The tunnel,
especially
cause
at
higher
irregular deflections
speeds
has
enough
in the wing causing
even at zero degrees angle of attack.)
turbulence
to
temporary non
The wing was
small
zero readings
then run through
angles of attack from 0 to 20 degrees at two degree increments.
We made
records of each incremental stop by strip chart records of the strain
gauge
readings,
still
photos
and
video
recordings.
Following
the
completion of tests at one airspeed we repeated the procedure at the next
higher airspeed.
Once a wing showed moderate
flutter,
angle of attack
was
Some of the wings will flutter at 16
not increased at that speed.
m/s at any angle of attack so that portion of the test for those wings
was omitted.
3.6 Flutter Boundary Tests.
A flutter boundary is
graph
attack
invariably
for
the
lower
a flutter
other is
a
a curve plotted on a airspeed vs. angle of
particular
airspeed
Previous
area.
boundaries of many of the wings in
experimenters
had
the
of
area
sweep.
flutter.
run
through
the
different
found the flutter
forward
zero the angle of attack
of
angles
When we encountered any flutter,
the
Our goal was to complete
our tests.
Our procedure was to set the airspeed,
then
graph,
while
tests for all the wings at both no sweep and 30 degrees
these
and
side
flutter free
a
is
side,
one
where
wing
attack
checking
for
either bending or torsion, we
made strip chart recordings and checked the
frequency with the strobe.
After finishing tests at that airspeed the angle of attack was reduced,
the airspeed was increased by 1 m/s and the test procedure was repeated.
When airspeed is increased to the point where the wing flutters at all
angles of attack or the maximum tunnel speed is
are taken and the test for that wing is
reached final readings
complete.
3.7 Flutter Test.
Our goal was to observe the actual shapes of the wing during both
In particular we wanted to
low and high angle of attack flutter.
concentrate on observing the wing tip, getting not only qualitative data
but actual measurements of the bending and twisting seen at the tip. We
chose 1 degrees AOA for the low angle of attack flutter and 10 degrees
AOA for the high angle of attack flutter.
Using data from the previously
run flutter boundary tests, we set the airspeed at the flutter speed and
then set the angle of attack.
small amplitude
we
increased
If
the flutter was intermittent or of very
the airspeed by as
much as one
meter per
second inorder to get a flutter motion visible on the video pictures.
Of
all the
with the various wings,
flutter cases
the wings fell into two
ones that had flutter dominated by torsion oscillations
main categories;
and ones that had flutter dominated by bending oscillations.
The torsion
flutter frequencies were in the area of 30 to 60 hz.
is much to
This
fast to see clearly with the eye and even using a video camera the motion
However since the motion is
will come out blurred on each frame.
in a periodic sense it can be captured by use of a strobe.
steady
The strobe
flash frequency is set near the flutter frequency by visual observation,
resulting in flutter motion that can be viewed at apparent slow motion.
The strobe has
The images also record relatively well on a video camera.
a very fast flash during which the wing moves very little, giving a sharp
elements,
camera
video
the
but brief image.
have a
however,
certain
amount of persistency, holding the image after the flash has stopped.
when the
video camera
scans
it
an image
has
nearly all
though the flash lasts less than 1/1000th of a second.
the
So
time even
For this reason
the video pictures made using the strobe provided all the quantitative
When the flutter-frequency was low,
data on the high frequency flutter.
as in
bending flutter,
and floodlight.
we took video pictures
Under floodlighting the
video picture
blank with an occational brief
when individual frames were viewed in
still
motion,
picture.
However,
the floodlight video
another matter.
When the strobe had flashed a single
clear,
it
was blank.
clear but
The strobe lighted video pictures were
pictures were easy to interpret.
otherwise
was
strobe
While recording under the strobe the
still to fast to take measurements.
video recording was
using alternately
video frame was
With the strobe flashing at about
5 hz
and the video camera scanning at 30 hz there is one clear frame followed
by approximately 6 blank frames.
This large spacing between good video
frames along with the slow flash rate in relation to random turbulence
induced motion
picture
video
cases.
of
in the tunnel made
it difficult to know when we had a
the wing at maximum deflection.
pictures
provided
the
bulk
of
Therefore
the data for
the
floodlight
the bending
flutter
In all cases,
down
of
the
wing
we took video pictures
under
both
strobe
and
from the side and looking
floodlight.
By
properly
positioning the mirror we could change from the side view of the wing to
the top view by merely changing the camera viewing angle. The side views
are used for qualitative evaluation of the flutter motion.
The side view
can identify nodes in the vibration shapes and is especially helpfull to
identify the presence of any second mode bending.
The top view clearly
showed the motion of the wingtip relative to the root.
Using the scale
on the background we were able to measure the deflection in both bending
and
torsion of both extremes
frequency
extremes
set
at
of the
approximately
flutter
twice
the
motion.
flutter
of the flutter motion could be captured
Similarly by using a double exposure on the still
With
the
frequency
on the
same
strobe
the
picture.
camera the two extremes
are caught on the same picture. This is shown in figure 3.6.
Figure 3.6
two
Double exposure showing extremes of flutter motion.
CHAPTER 4.
RESULTS
4.1 Static Deflection
Perhaps
deflection
the
best
way
to
Picking the
is
is
looking
by
While
the
[02/90]s
property
at
two
that
gives
the
wings
the
layup wing only
and only twists under a moment load,
this
static
layup wings
[+152/0]s
the
[+152/0]s
load or a moment
layup wing both bends and twists under either a force
It
the
The most striking thing we see is
effect of bending-torsion coupling.
load.
of
and B.2, we can compare the deflections
produced under similar loading.
under a force
results
and the
[02/90]s
results as shown in figures B.1
bends
the
and experiment
analysis
Rayleigh-Ritz
typical examples.
examine
their
interesting
aeroelastic properties.
Some of the noteworthy properties of all the wings are:
1)
absolute
of
value
the
balanced and unbalanced
of
the
outer
with
for
plies
increased
both
the
fiber
angle
layups
(balanced means that for every +0 ply in
a -8 ply).
the wing there is
2)
increases
Bending-torsion. coupling
The
bending-torsion
coupling
for a
[+0/O]s
layup
is
angle
is
opposite in sign from a [-0/Os layup, but equal in magnitude.
3)
Bending
stiffness
decreases
as
ply
fiber
increased.
4) Torsion stiffness increases with increases in ply fiber
angle up to 30 degrees.
The 45 degree ply fiber angle layup reverses the
trend and has a slightly lower torsional stiffness than the 30 degree ply
fiber angle layup.
With a few exceptions all wings deformed in close agreement with
linear theory in
the
linear
degrees.
the so called small deflection range.
range was
At
larger
bending up
deflections
to 5 centimeters
the wings
become stiffer than linear theory predicts.
and
For these wings
twist up
show a general
to
tendency
6
to
This was an expected result
due to the increasing influence of non-linear factors in the wing and in
the test procedure.
For example, loads were always applied perpendicular
to the wing undeflected position (force in the z direction and moments
But with large deflections the loads were no longer
about the x axis).
at right angles
to
the wings new position.
Therefore
the
amount of
effective load was somewhat less that the actual applied load.
The coupling deflections, that is,
the twist due to applied force
and the bending due to applied moment, held close to linear theory on the
wings
that had
a large
amount of
coupling.
The
exceptions
coupling deflections of bending due to moment for some of
coupled wings.
In
the
were
the
lightly
particular, the two balanced layup wings with the
[±30/0]s layup wings.
[-+15/0]s and
lowest bending-torsion coupling, the
Both of these wings had large deviations from linear theory with bending
due
to
expected
moment in
one
bending
direction.
results
examining the [±15/ 0 ]s
in
the
other
E 88
direction.
E8
-
In
produced
particular,
F 2.880
R
C 8.88
E
R
C 8.88
E
N-2.88
-2.88
R-R Analysis
Experiment
O
4.88
O/
F 2. 88
0
0
-4. 0
-4.4.1 A I 1
-1.88
E-01
8.088
W(CM)
1.88
E 81
2.88
O
1
-1.88
1
.88
E-
-T
-2.88 -1.88
1
1.88
8.88
(RAD)
2.88
M
0
M
E
N 8.88
E-81
0
O
N
N
M-2.88
figure 4.1
sign
opposite
layup wing test results, shown in figure 4.1,
4.08
M
0
M
E
N
T
of
Moments
-2.88
0.88
S(<CM)
1.88
2.68
E 88
-2.88 0.88 2.88
~C (RAD)
E-
static deflection test results for [±15/0]s layup wing.
we
see that when a positive moment was applied the wing bent slightly in the
Yet ,
direction opposite of the one predicted by theory.
moment the wing bent as predicted.
found
in
that
its
Upon close examination of the wing we
state
unloaded
for a negative
it
had
a
slight
as shown in
negative looking down the wing from the wingtip,
camber,
chordwise
figure 4.2.
a certain minimum load must be applied in
According to plate theory,
the
wing
Figure 4.2 Initial camber (exagerated) in the [(15/0]s layup wing.
negative direction to buckle the plate in
the negative direction.
Since the coupling force is
moment the expected bending is
only 0.3N)
it
apparently
and therefore .the
order to get any deflection in
is
(at 0.2 Nm of
weak
equivalent to that produced by a force of
unable
to
"pop out" the
wing does not deflect in
camber
in
the wing
the positive direction.
The
[±30/0]s wing had a similar problem although somewhat less severe.
The
orienting
initial
the
plies
camber
exactly
in
at
the
the
wings
proper
them to be cured slightly out of alignment.
to
avoid,
and
very
small deviations
will
was
probably
angle
during
This is
produce
caused
by
not
layup,
causing
not an easy problem
significant
warping.
Nearly all wings had some inadvertent warp but the large coupling present
in
most wings could develop enough force to overcome the problem.
It was
the lightly coupled wings that initial camber caused significant
only in
deviation from linear theory.
One point to note
the
camber and
is
that if
we bent the wing enough to pop out
then applied the moment, coupling behavior was
normal.
the wind tunnel the lift is strong enough to
When airloads are applied in
pop out the camber so the wing could still
be expected to act according
to the linear analysis.
4.2 Steady Airload Deflection
Qualitatively,
one would
coupling
than
both
logically expect.
(positive bending
their counterparts
indicates
and the
wings with
The
produces negative
test results were what
negative bending-torsion
twist) deflected much less
bending-torsion coupling.
with positive
This
that the negative coupled wing did reduce and redistribute the
by
airload
the analysis
decreasing
the
of attack of
angle
the wing
sections.
The
video pictures confirmed the decreased angle of attack.of the wing tip.
However,
wing,
we
if
we compare the [+152/01s
see
that
the
wing
with
layup wing with the [+30 2 /0]s
the
more
negative
layup
bending-torsion
coupling deflects more than the one with the less negative coupling.
The
reason is that as we increase the ply fiber angle we not only increase
bending-torsion
the
also decrease
coupling but we
bending
stiffness.
Therefore, although the [+302/0]s layup wing may twist more and therefore
lower the airload more than the
[+15
2 /0]s
layup wing the difference in
bending stiffness offsets the difference in coupling.
We
can
see
the
interplay
between
bending
stiffness
and
bending-torsion coupling if we examine the steady airload deflection at
one airspeed and no sweep
particular degree
ply
calculations results
(A = 0)
fiber
for
angle.
for the four
11.5 meters per second airspeed.
the four wing
Figure
4.3
layups
shows
15 degree fiber angle
The twist angles
the
that use a
theoretical
layup wings at
follow directly the
amount of bending-torsion coupling. The bending deflection (w) however,
depends on the combined effects of bending stiffness and bending-torsion
coupling.
The
(115/0]s layup
bends
wing
6
the
least
because
of
its
12
STEADY AIRLOAD
ANALYTICAL RESULTS
4
1
[+152/0]s
2
115/0]
3
[±is/o]s
4
[-15/01 s
4
s
8
U
4
1
3
2
3
3
0
_
0
_
0
_
4
_
8
_-4
12
16
20
0
4
8
ROOT AOA (DEG)
Figure 4.3
12
16
20
ROOT AOA (DEG)
Steady airload analysis on the four wings with a 15 degree
ply fiber angle at 11.5 m/s and A = 0.
The [+152/0]s bends
higher bending stiffness and slight negative twist.
just slightly more. Apparently its negative coupling could not completely
The following two wings
compensate for the decreased bending stiffness.
follow in
order of
angles of attack
Then at higher
less
than the
bending-torsion coupling at
[+15
2
/0]s
the
At a root angle
layup wing.
the slightly higher angle of
results
in
a
very
minor
attack of
increase
in
at
a
fixed
root
angle
coupling will cause a wing to both
of
the
the
Therefore, the higher wing stiffness dominates.
that when
of
attack.
layup wing bends a little
[T15/0]s
degrees
only
low angles
attack
of
attack of
[T15/0]s
coefficient
20
layup wing
of
force.
In general we can say
negative
bending-torsion
lessen the total airload and also
redistribute the airload toward the root, while positive bending-torsion
coupling will cause a wing to increase the total load and redistribute it
toward the wing tip.
.
One point worth noting concerning the
[+15/0]s layup wing is the
that
fact
it
under airload
force
coupling
had
twist.
little
In
case
the
cancel
the
this
enough
strong
just
approximately
was
to
In twisting this particular
the airload.
generated by
twisting force
very
It is easy to imagine a case where
wing acted much like a rigid wing.
this could be an advantage in a wing design.
the
higher
airspeeds
coupled wings
showed
a tendency
At
the
in
even
that produce
a coefficient
the
For example.
of
force
the highly
test
at low
root angles
of
positive
attack to
near the tip reached angles
twisting until the wing sections
continue
force.
used
near
the maximum
coefficient
of
layup wing at 11.5 m/s at 4 degrees
[-452/0]s
root angle of attack had twisted an additional 6 degrees and reached a
6
of
deflection
of
coefficient
put
there
of
percent
84
sense we
divergence
classical
at
tip
the
In a practical
force.
Unlike
diverged.
This
cm.
maximum
can say the wing has
limits
are
to the
twist
because we have used a realistic coefficient of force that has a maximum
value.
Comparing the analytical and experimental data quantitatively (see
figures
through
C.1
than those
had experimental
layup wing at 16 m/s
[±15/0]s
In particular
values
much higher
high experimental
The
analysis.
by the
predicted
considering
satisfactory
[02/90]s layup wing at 11.5 and 16 m/s and
at higher angle of attack the
the
is
agreement
measurement except for a few areas.
accuracy in
atainable
the
C.13)
readings
were coincidental with the appearance of torsional flutter of the wings.
In
other words once the wing started to flutter in
This is a preliminary indication
more than predicted by the analysis.
that
when
in
the
flutter,
torsional
also bent
torsion it
wings
a
have
higher
average
coefficient of force at a certain airspeed and angle of attack than they
would without the flutter.
both
increase
flutters.
inorder
to
vibrations
or
redistributed
are
wings
Clearly
the
sustain
the
cause
increasing the
some
drag.
This means that either the lift
are
These
more
taking
vibrations.
additional
toward
Also
it
disturbance
reasons
for
the
tip
when
out of
energy
seems
to
or drag or
logical
the
the
wing
the
air
that
the
airstream
thus
the increased deflection are
speculation, however,
the
would have were
it
steady
analysis
generated
by
the
conclusion we
can draw is:
Once a wing
does not vibrate about the steady state position it
starts to flutter it
airload
one
not fluttering.
Therefore we also conclude that any
that does
unsteady
airloads
not
include
the
in
flutter
will
of
average
forces
underestimate
the
average airload and therefore underpredict the average deflection.
4.3 Divergence
The
Velocities
can be considered the
investigation
divergence
limit of the
We are looking for the static stability
steady airload investigation.
linear feedback system mentioned in chapter
limit, the point where the
two no longer converges.
The investigation pointed out the extreme differences between the
due
wings
examining
table
4.1
ply
different
to
solely
we
see
that
fiber
angle
somewhere
that of
below
On
wing.
coupled
For a straight wing (A=O)
The actual value of the crtitical
coupling the wing will not diverge.
lightly
ranges
With a sufficient amount of negative bending-torsion
the rule is simple:
is
By
patterns.
for a straight wing divergence
from a low of 12.4 m/s to a high of infinity.
amount
layup
the
the
[-+15/0]s
hand,
other
our
layup wing,
positive
most
bending-torsion
coupling lowers the divergence speed in relation to its magnitude.
For
the 30 degree swept forward wing
there
showed
is
an
divergence speed.
due to geometry,
optimum
fiber angle
(A=-30) the investigation
layup pattern
for
increasing
Because bending causes increases in angle of attack
all the wings have a finite divergence speed.
However,
by using the optimum layup pattern the divergence speed can be more than
doubled over that of the uncoupled layup wing.
The optimum layup pattern
among those we used was the [+152/0]s pattern.
By examining the [02/90]s
and
the
[+302/0]s
wing
layup
results
we can
speculate
that around
a
[+20 2 /0]s layup wing would have the highest divergence speed.
Of all the analysis
gave
results
that
best
done in
agreed
this project the divergence
with
experimental
results.
analysis
This
can
WING
A = 0 degrees
A = -30 degree
Theoretical
Experimental
Theoretical
Experimental
(m/s)
(m/s)
(m/s)
(m/s)
[02/90] s
[+1 52/01s
[± 15/0] s
31.4
******
[F 15/0]s
24.2
20.9
19
[-1 52/0] s
16.5
15.6
15
[+302/0] s
39.4
[±+30/0] s
26.1
[T30/01 s
24.7
18.5
[-302/0] s
14.2
12.4
[+452/0 s
[±+45/0 s
18.4
[ 45/0] s
24.1
14.9
[-452/0] s
14.0
11.3
Table 4.1
Wing divergence speeds.
probably be attributed to the use of three dimensional aerodynamics.
For
the experiments, the wings were said to have diverged when at near zero
degrees root angle of attack, the wing bent to a large deflection.
times
this
deflection was
bending frequency.
oscillatory
range
but
fluttered.
the
divergence
before
low frequency
below first
This oscillation was due to non linear effects and
was beyond the scope of our analysis.
layup wing,
at a
Many
reaching
For a few wings,
velocity was
that
speed
within
in
the
the
like the [02/90]s
wind
tunnel
experiment
the
speed
wing
Therefore those divergence speeds, along with the ones higher
than maximum tunnel speed, were not verified by experiment.
4.4 Natural Frequencies
from the initial deflection
Ascertaining the natural frequencies
tests
is
a skill that requires some
fine discrimination.
Due
to the
nature of composite material, vibrations are damped in relation to their
frequency.
For some of the wings the second bending and first torsion
vibrations were detectable on the strip chart for only a fraction of a
second.
chart
The fact that all three
and
frequencies
complicated
some
for
are
the
not
wings
very
problem.
the
frequencies
second
a
and
bending
different
As
are present on
result
and
are
for
two
the
highly
wings
the same
first
torsion
coupled
we
ascertain the second bending frequency and for all wings
could
also
not
the measured
second bending and first torsion frequencies are no more accurate than
within about 2 hertz.
First Bending
First Torsion
Second Bending
Theo.
Exp.
Theo.
Exp.
Theo.
Exp.
(hz)
(hz)
(hz)
(hz)
(hz)
(hz)
[02/901]s
10.8
10.5
[+15/0]s
8.5
[±15/0]s
9.9
[+302/0]s
6.0
7.1
[±30/01 s
7.8
8.0
[+452/0]s
4.6
6.2
[±45/01 s
5.7
6.0
WING
_
9.2
10.0
Table 4.2
Wing natural frequencies.
The results of the test along with the analytic results are shown
in
table
4.2.
As
expected,
since
all
wings
the
have
same
mass,
the
bending and torsion frequencies are directly related to the bending and
Comparing the calculated to the measured frequencies,
torsion stiffness.
we have generally good agreement. For the three highly coupled wings, the
[+30 2 /0]s
[+152/0]s,
and [+452/0]s,
bending frequency
the measured first
was higher than calculated, while the measured first torsion frequency of
the [+45 2 /0]s
was lower than calculated.
These differences are probably
due to the bending-torsion coupling in the wings)
vibration were present at the same time in
since both modes of
the test.
4.5 Flutter Velocities
The data from the flutter boundary
D.
tests is
plotted in
Appendix
Because of the way the boundaries are graphed we can see how the root
angle
of
the flutter speed.
attack changes
From this table we can see
table D.1.
tests are shown in
The results of the flutter
the twist of
the wing tip during flutter and therefore can calculate the tip angle of
attack as well.
For
those
wings
before they diverge,
change in
that
at
zero
degrees
angle
of
attack
flutter
a few general statements can be made concerning the
The type of
flutter characteristics with root angle of attack.
flutter changes from a bending-torsion to more of a pure torsion flutter
as
angle
of
attack
is
increased.
This
can be
frequency increasing toward the first torsion
amount of bending in the flutter.
For example,
the
[02/90]s
seen
frequency
by
the
and
flutter
a smaller
Also, the flutter airspeed decreases.
layup wing unswept went from 33 hz
to 40 hz,
from a Aw of 0.7 cm to 0 cm,
and from an airspeed of 26 m/s to 13 m/s at
10 degrees angle of attack.
These same trends were
previous stall flutter work in reference 14.
noted by Rainey in
However as the wings become
coupled in bending and torsion these trends are changed.
In the case of
the [+302/0]s layup wing there is no frequency change and the airspeed is
nearly the same for both 0 and 12 degrees angle of attack flutter.
If we
examine the wing conditions we see that at 0 degrees angle of attack the
twist is
average
+2.4 degrees,
average twist is -4.8 degrees.
angle of attack the
while at 12 degrees
Thus the tip is at an increased angle of
attack of only about 4.8 degrees versus 12 degrees for
This
the root.
could account for the similarity in flutter behavior at the two different
root angles of attack.
The wings that diverged at 0 degrees root angle of attack before
angle
attack
of
Generally
increased.
was
flutter
behavior
flutter
speed
bending
similar
exhibited
flutter all
they would
the
as
would
the frequency would increase and the flutter would become more
decrease,
mild, making it more like a bending flutter and less like divergence.
The wings
were the
the best at low angle
that resisted flutter
speeds were beyond
In
[±45/0]s layup wings.
and the
[+30/0]s
their flutter
However,
tunnel.
the wind
the maximum speed of
fact,
of attack
as
angle of attack was increased their flutter speeds dropped below that of
their unbalanced
the
[+302/0]s
[+452/0]s
the
and
layup
Due to the light bending-torsion coupling of the balanced layup
wings.
wings
counterparts,
the
they exhibited
including
decrease
the
normal
in
flutter
flutter speed
as
about
talked
trends
root
angle
of
earlier
attack
is
Among the wings with an unbalanced laminate layup pattern a
increased.
ply fiber angle of 30 degrees seems to give the highest flutter speed.
When the wings were swept forward the addition to angle of attack
Tfe [02/90]s layup wing
made by bending had a strong effect on flutter.
lower airspeed
at 0 degrees angle of attack fluttered at a
unswept,
while at higher than about 8 degrees angle of attack it
from a bending torsion flutter to a bending flutter.
layup
[+302/0]s
increase
in
decrease in
wings'
tip angle
attack.
oscillation.
bending-torsion
of
coupling
attack due to bending
flutter speed with an increase in
[+452/0]s layup wing,
of
than when
The
being weak in
frequency
shown
bending,
in
table
Note that the average bending is
The [+152/01s
could
so
changed
they
not
offset
and
the
also showed a
root angle of attack.
The
now diverged at all angles
D.2
is
nearly
a post divergence
17 cm!
The wings
that had diverged at 0 degrees sweep also diverged at -30 degrees sweep,
the speeds were just slightly lower.
The differences in the low angle of attack behavior of the various
In order to examine
wings was also evident in the theoretical analysis.
a typical cross section of the wings I have taken the graphs for the four
wings with 15 degree fiber ply angles from appendix D and reproduced them
the torsional branch will go unstable at 24 m/s
method predicts
frequency around 27 hz.
change
in
approached the torsional branch has a
Looking at the other
frequency.
bending-torsion
flutter,
the
to the
[+152/0]s
layup wing.
similar
in the
stable
but
stability
then
while
The
divergence.
[-15 2 /0]s
layup
divergence speeds
because
layup wing also goes
this
at
the
goes
main
wing
difference
a
has
between
slightly
the
lower
goes
This
zero.
to
both go
initially very
is
branch
critical velocity it rapidly
frequency
the
airspeed.
layup wings
[-152/0]s
The
first bending branch.
very
see behavior
we
[+15/0]s
The
the [P15/0]s and the
The next two wings,
unstable
layup wing,
[±15/0]s
wing with negative
the torsional branch, just at a slightly higher
unstable in
a
at
25 m/s and 32 hz.
values were
The experimental
Note that as flutter velocity is
large
the V-g
[+152/0]s layup wing first,
Looking at the
figure 4.4.
here in
two
to
shows
wings
divergence
neutral
classical
is
that
the
The
speed.
indicated here are lower than the experimental values
analysis
not
does
consider
finite
in
the
the divergence
and
effects
span
aerodynamics.
For 0 degrees angle
V-g flutter analysis
figure
4.5.
of attack the results of
along with experimental
The unbalanced
separate graphs.
and balanced
data points
are
shown in
wings
are
shown
laminate
on
For
The [02/901s layup wing is plotted on both graphs.
completeness the V-g flutter analysis for the wings at 30 degrees swept
back is included.
There
is
no experimental
data for
30 degrees
swept
back.
Let's
look
at
the
unbalanced
unswept wing if the fiber angle is
the limiting factor otherwise
laminate
wings
first.
below about -5 degrees
the wing will flutter first.
For
the
divergence is
Despite the
[15/1Os
2B
11
0
20
40
VELOCITY (M/S)
60
20
0.4
40
VELOCITY (M/S)
6(
0.4
-1
IT
0
40
1T
0
60
40
60
2B
-0.4
2B
-0.4
1B
80
80 r
[F15/0]s
1B
[-152/0]s
2B
1T
40
0
40
1B
I
It
20
0.4
2B
20
40
VELOCITY (M/S)
40
VELOCITY (M/S)
60
0.4
1T
0
60
-040
2B
-0.4
-0.4
B
Figure 4.4 Flutter analysis diagrams for the four wings with a 15 degree
ply fiber angle.
large
variations
in
stiffness
and
bending-torsion
coupling
from
the
[0 2 /901s to the [+45 2 /0]s layup wings, flutter speeds were all within 5
m/s for the four wings.
A ply fiber angle of +30 degrees seems to give
the highest flutter speed both theoretically and experimentally.
For 30
degrees swept forward, due to the effect of bending on angle of attack,
divergence dominates for all ply fiber angles
from
[15
2
about
/0]s
+5 to
+30
we
could
the available speed range of the wind tunnel.
to note that the limiting speed
degrees
particular,
swept
region of
not
the
get
layup wings to flutter or diverge at low angle of
and [+302/0]s
attack in
In
degrees.
except in the
higher
forward is
It
is
worthwhile
(either flutter or divergence) for
than
for the unswept wing
30
for fiber
angles from -5 to +35 degrees according to the theoretical calculations
was true for fiber angles of +15 and +30 degrees.
and experimentally it
The
layup wing went into torsional flutter with a large
(0 2 /90]s
average bend at an airspeed below its calculated divergence speed.
was
not predicted
bending
by
deflection
the V-g analysis.
the
was
wing
Because
actually
in
of the
a high
large
angle
is
of
attack
high angle of attack flutter for the [02/90]s layup wing
Also this type of flutter is
lower than when at low angle of attack.
not covered by our analysis.
we see that when a wing is
has
average
As we have
flutter despite the root angle of attack being only 1 degree.
seen previously,
This
Looking back at the steady airload analysis
it
below divergence speed,
close to, yet still
large deflection even at low angle of attack.
In the swept forward
case bending causes increases in angle of attack so we can see how when
nearing divergence speed even with a low root angle of attack we can get
a high tip angle of attack bringing with it the possibility of flutter.
Switching now to the balanced laminate wings, for the unswept case
we have divergence for outer layer
flutter elsewhere.
fiber angles below -10
tunnel
maximum
and
These wings had the highest predicted flutter speed
at a outer ply fiber angle of +45 degrees.
the
degrees
speed
so
we
could
This speed was higher
not
determine
its
than
accuracy.
Experimental data was reasonable except for the [P15/01s layup wing which
will be discussed in the next paragraph.
VELOCITY (M/S)
60 -
FLUTTER DIVERGENCE
THEORETICAL ANALYSIS
A
A = 0 DEGREES DATA
O
*
_A = -30 DEGREES DATA
A= +30 -40 +
I
I"
I
=-+30
-
TUNNEL SPEED LIMIT
~A= 0
DA
A= -30
n2 -3
*
-30
-=
L
-30
0
-15
PLY FIBER ANGLE (DEG)
VELOCITY (M/S)
60 T
FLUTTER
DIVERGENCE
THEORETICAL ANALYSIS
= 0 DEGREES DATA
= -30 DEGREES DATA
S
A = +30 -
S=
+30
40 -1A= 0
TUNNEL SPEED LIMIT
A=
-30
/1 -30 *
I
T45
'
+30
'
-
:15
'
0
'
±30
PLY FIBER ANGLE (DEG)
Figure 4.5
Effects of fiber ply angle on flutter and divergence speeds.
For
divergence
30 degrees
the
was
forward
limiting
for the
sweep
instability
balanced
laminate wings,
the
throughout
fiber
angle
We see a calculated best outer fiber angle of about +20 degrees.
range.
Experimental data was close to calculated with the following exceptions;
the
[02/90]s,
[±30/0]s
[±15/0]s,
at
[;15/0]s layup wing from the unswept wing case.
The
and
sweep
forward
degrees
30
the
[02/90]s layup wing
was discussed previously and in fact what was said for the [02/90]s layup
wing also
applies to all
Their
these wings.
torsional
all
cases are
flutter characterized by high average bending and speeds below predicted
suggesting a high angle of attack flutter at the wing
divergence speed,
By checking
tip.
for
the
[+30/0]s
layup wing
(wavg) in
bending deflection
the average
(19.8 cm),
table D.2
one could argue that in
fact the
wing has diverged and the calculated divergence speed is just too high.
However
I
present
the
argument
enough to allow for high angle
that
of
before
attack
the wing diverged
Once
flutter.
fluttering, the increase in average force caused by
wing
to
bend
average bend.
even
more
(see
discussion
section
the
limit and it
is
4.2),
at a airspeed
airspeed predicted by the theoretical analysis.
bent
wing
is
flutter caused the
thus
Whatever your point of view, one thing for sure is
have reached a stability
it
the
high
that we
lower than
the
CHAPTER 5. CONCLUSIONS AND RECOMENDATIONS
Having now investigated some of the many varied and interresting
properties demonstrated by the graphite/epoxy composite material wings I
will finish by pointing out a few of the
conclusions we can make and
offer some recomendations for follow up work.
The wings showed very near linear bending and twisting behavior up
to large deflections
in the
range in which we can
apply our
linear
gives a
This
tests.
static
theory.
large useful
However,
on
bending-torsion coupled wings small imperfections in the wings
warp
can
cause
large
deviations
from
linear
theory
lightly
such as
concerning
the
bending-torsion stiffness coupling reaction.
The
steady airload analysis
gave good results
angle of attack as long as the wing did not flutter.
of a realistic
lift
curve,
rather than a linear one,
up to
20 degrees
Certainly the use
contributed to the
accuracy of the results.
Once the wings did flutter they had an average
deflection
the
greater
than
steady
airload
theory
predicted.
This
limited the useful range of the steady airload analysis and suggests the
usefulness
of
investigation
into
the
resultant
average
forces
present
during flutter, both lift and drag.
The initial
deflection frequency
tests were a quick easy way
to
get approximate values for the natural frequencies.
The divergence and
flutter investigation
once
again showed
the
large variation in wing aeroelastic properties caused by changes in ply
fiber angle.
Straight wings can be made divergence free, while by proper
selection of fiber angle, even a swept forward wing can have a divergence
speed higher than a similar wing without bending-torsion coupling at no
sweep.
By using three dimensional aerodynamics the divergence speeds of
the wings
wings.
were
closly
The flutter
accurate.
We
were
predicted
calculations
unable
to
for both
straight
and
forward
for the straight wings were
check
the
accuracy
of
the
swept
resonably
flutter
calculations for the forward swept wings because of the wind tunnel's low
maximum speed.
However, by extrapolating to
low angles of attack the
experimental data we were able to obtain at high and moderate angles of
attack,
we
can
predict
a
calculated flutter speed.
speed
flutter
close
to
the
theoretically
We did discover that the lightly coupled wings
had a tendency to go into a torsional flutter,
characterized by a high
average bend and therefore a high angle of attack at the wing tip, when
approaching but still
below predicted divergence speed.
deserves further investigation.
This phenomenon
A stall or high angle of attack, flutter
analysis might give some insight here.
Finally,
a good follow on investigation would be to make and test
some built-up wings versus the cantilevered plates used here.
wing with a typical box-beam or honeycomb
A built-up
construction would smooth out
any initial warp or twist in the laminate skin, thus eliminating one of
the
problems
airfoil
and
we
is
had.
the
That type wing would be
next
logical
step
graphite/epoxy composite material wing.
toward
closer
building
to a pratical
a
functional
REFERENCES AND BIBLIOGRAPHY
Ricketts,
R.H. and Weisshaar, T.A.,
1. Hertz,
T.J.;
Practical
Forward-Swept Wings",
Astronautics and
"On the. Track of
Aeronautics
Magazine,
January 1982.
2. Hollowell, S.J., "Aeroelastic Flutter and Divergence of Graphite/Epoxy
Cantilevered
Plates
with
Stiffness
Bending-Torsion
Coupling",
M.S.
Thesis, Department of Aeronautics and Astronautics, M.I.T., January 1981.
3.Jensen, D.W.,
"Natural Vibrations of Cantiliver Graphite/Epoxy Plates
with Bending-Torsion Coupling", M.S. Thesis, Department of
Aeronautics
and Astronautics, M.I.T., August 1981.
"Aeroelastic Flutter and Divergence of Rectangular Wings
4. Selby, H.P.,
with Bending-Torsion Coupling", M.S. Thesis, Department of
Aeronautics
and Astronautics, M.I.T., January 1982.
5. Hollowell, S.J. and Dugundji, J.,
of
Stiffness
Graphite/Epoxy,
Coupled,
AIAA/ASME/ASCE/AHS
"Aeroelastic Flutter and Divergence
Structures,
Cantilevered
Structural
Dynamics
Plates",
and
23rd
Materials
Conference, New Orleans, La., May 10-12, 1982, AIAA Paper 82-0722.
6. Tsai,
S.W. and Hahn, H.T.,
"Introduction
to
Composite
Materials",
Technomic Publishing Co. Inc. 1980.
7. Bisplinghoff,
R.J.;
Ashley,
H. and Halfman,
R.L.
"Aeroelasticity",
Addison-Wesley Publishing Co. 1955.
8. Bisplinghoff, R.J. and
Ashley, H.,
Dover Publications, Inc. 1962.
"Principals of
Aeroelasticity",
9.
Dowell,
E.H.,
"A Modern
Course
in
Aeroelasticity",
Sijthoff
and
Noordhoff 1980.
F.W.
10. Riegels,
"Aerofoil
Sections",
Butterworths
Publishing
House,
1961.
11.
DeYoung, J.
and Harper, C.W., "Theoretical Symetrical Span Loading at
Subsonic Speeds for Wings Having Arbitrary Planforms", NACA Report 921,
1948.
12.
Spielberg,
I.N.,
"The
Two-Dimentional
Incompressible
Aerodynamic
Coefficients For Oscillatory Changes is Airfoil Camber", Journal of the
Aeronautical Sciences, June 1953.
13.
Weisshaar,
T.A.
"Divergence
of
Forward
Swept
Composite
Wings",
Journal of Aircraft, June 1980.
14.
Rainey,
A.G.,
"Preliminary Study
of Some Factors
Which Affect
the
Stall-Flutter Characteristics of Thin Wings", NACA TN 3622, March 1956.
APPENDIX A
MODE SHAPE INTEGRALS
AND
THEIR NUMERICAL VALUES
2
d .
using
and
dx
i
i
2
(
I
II
dx = 1.000
1
0
(
1
I2
2
dx = 1.000
2
0
2
13
adx = 0.500
I ( *
=
0
2
1
(
I4 =
4
)
dx
=
0.500
0
2
I(
5I
=
55
5)
dx = .03333
0
12.36
S2
16
17
=
£3
)
2
I
0
dx=
12.36
3' dx = 3.744
=
dx 2
I8 =
2
f
' dx = 3.249
1 "I
0
9
=
f
0
45
1
dx = 0.4340
2
10
dx = 485.7
)
2'
2
0
2
11=
2
,
dx = -6.698
3 '3'
'
0
I12
13
2
£2 J
0
5
2
f
dx = 63.55
4'4
2
dx = -2.868
0
2
f
114
)
3
dx = 3.044
0
£
)
115
0
2
dx = 1.234
2
I16 =
dx = 0.4293
1
0
= 0.1739
5 dx
3
f
0
117
2
f
18
)
4
2R
I 1Q
=zf(
2
2
dx = 11.10
4'
dx = -6.712
5'
4'
f
20
dx = 246.6
0
I 21
4' 9dx
= -0.3023
0
2
22 =
f
0
)
*5
£
5
23
)
dx = 4.000
2
dx = 0.3334
0
It24
f
0
1
dx
2.000
£
2
kJ (
125
'
)
dx = 4.6478
0
26
f
dx = -4.7594
'
0
02 dx = 0.7595
1
f
I27
0
x
28
f
2 ' dx = -7.3797
1
0
1
129
dx = 0.67786
=
0
3 dx = 1.5173
30= f
0
I31
31
1
a
1
4
4 dx = 0.146305
1
I32 =
4 dx = -0.19609
0
1
33
33
fI 1
1
0
5
dx = 0.12344
1f
134
92 dx = 2.000
2
f
0
I35
5 dx = 0.35466
2
36
I
2
)
dx = 32.4163
1
37
I
=
2
)3 dx = 0.1936
0
I
39
=
1
1
3 dx = -2.7155
2
f
0
I38
f
2
dx = 0.61191
4 dx = 2.8612
2'
40 =I
4
0
1
41
I42
Y.
I
0
2
2
5
dx = 0.13016
dx = -0.2524
1
I43 =
f 43 15 dx = 0.11074
0
I44 =
fI4
0
5 dx = 0.06414
APPENDIX B
STATIC DEFLECTION
INVESTIGATION RESULTS
-
E N0
R-R Analysis
Experiment
0
4.09
4.00
F 2.00
0
R
C 0.00
E
0.00
H -2. 00
N-2.00
S
-4.00
1.00
0.00
W CM
-1.00
E-01
2.90
E 01
-1.00 -Q.58 8.09 8.58
AHG RAC
E-01
0
0
0
00
1.00
E-81
0
EI 0.09
T
SOO
0
N
o0
-2.90
-2.09
O
•
ON
• •
.
-1.09
.
.
I a--- a-
8.0
1 (C1)
Figure B.1
-
I
O
0
0
A
-2.00
1.00
E GO
[02/90]s layup wing, static deflection.
8.88
NG (RAOD)
E-81
E 00
4. W
E 00
4.00
O
R-R Analysis
Experiment
2.00
0.88
8.00
N-2. 8
-4.00
-2.80
6-
-1.00
E-81
0.00
W (CM)
1.00
-
E 1
-4.80
-4 .800 -2.00 6.88 2.00
E-01
ANG (RAD)
4.880
4.00
E-81
2.00
0
M
E
N 0.88
T
S2.00
0
M
E
N 0.80
T
N
-2.88
H-2.80
*.mu~amaa
-4.880
00'
-O
-2.00
Ml
8.00
W(CM)
2.80
4.00
E 88
-4.0 1
1a ll
I
l
i
-4.00 -2.80 08.8
2.00
ANG (RAD)
Figure B.2 [+152/0]s layup wing, static deflection.
ll
4. 00
E-al
E 00
E gO
4.00
4.00
2.00
2.00
0.00
0.00
N-2.00
N -2.00
-4.00
-4.00
0.0
W(CM)
-1.0088
E-01
1.00
-1.80
E 01
2.88
0.00
ANG (RAD)
E-01
-2.00 8.00 2.08
ANG (RAD)
E-91
E-1
2.00
M
0
M
E
N 6 .00
T
0
M
E
N 9.00
T
N
M-2 .00
N
-2.00
oi
m Ia
-2.00 -1.00
Figure B.3
O
R-R Analysis
Experiment
0.00
I (CM)
t Ia 1i mI
1.00
2.00
E 00
[±15/0]s layup wing, static deflection.
E 06
R-R Analysis
Experiment
O
2. 6
00
8.0
-2.00
-0
-2.00
!o
-1.00
0.00
E 1
W(aCM)
E-01
2.06 4.00
-4.06 -2.66 0.06
E-81
ANG (RAD)
E-1
E-01
1.00
2.00
2.00
1.00
0 1.00
M
0.00
N 0.00
T
E
M
-2. 060
H-2.66
00
.
-2.00 -m
0a
HI i
-6.00 -3.00
Figure B.4
0.00
W(CM)
3.00
6.00
E OB
II .
a.
a.
-4.00
I
.
a
a
.
.
0.00
ANG (RAD)
[+30 2 /0]s layup wing, static deflection.
a.
I.
a.
4.80
E-BI1
E 0
E0
2.008
R-R Analysis
Experiment
2.00
F
0
R
C 0.00
E
0.00
N
-2.00
-2.00
Ol
I I I I
-1.00
E-61
i i i I I
0.00
W(C)
m
-2.80 -1.00
0.00 1.00
E-1
ANG (RAO)
1.00
E 01
2.00
0
i e.00
T
H
M-2.00
-2.00 -1.00
N
M-2.06
0.00
M (CM)
1.8
2.00
E 0G
-2.0
Figure B.5 [±30/0]s layup wing, static deflection.
80.00
ANG (RAD)
2.00
2.00
E-01
E 00
E 00
R-R Analysis
Experiment
0
1.00
0
0.00
-1
- O
-1.00
_
a .. I_ I_ I_ A
_ I_
-1.00
0.00
W (CM)
E-01
I *
,.,
E
M
-2.00
* lI
4.00
E-01
M
0 1.06
M
E
0 0
N 0.00
T
N -1.66
"
t
-2.00
-6.00 -3.00
Figure B.6
I
2.00
M
0 1.00
-0
E
-N 0.00
T
-1 .00 ,u
I1 I , 1
-4.00 -2.00 0.00 2.88
ANG (RAD)
E-01
1.66
2.00 D
M
I I
0.00
M (CM)
3.00
6.8
E 0B
-4.00 -2.00
0.00
ANG (RAD)
[+45 2 /0]s layup wing, static deflection.
2.08
4.00
E-1
E O
2.00
F
..
0
R
C 8.ee
E
.ee
N- .80
N-1 .0e
-2.00
-1.00
E-01
0.80
W(CM)
-2.00
1.80
E 81
E-1
1
0
M
0
M
N 0.80
T
1.0E
E-81
2.80
N
N
M-2. 800
Figure B.7
8.00
ANG (RD)
E
N
T
E
-2.80
-1.00
mb
-2.80
WO
iIIo
-1.80
0.80
U (CM)
1.80
2.00
E 0G
-2.80
-1.00 8.80
1.80
ANG (RAD)
[±45/0]s layup wing, static deflection.
2.08
E-81
APPENDIX C
STEADY AIRLOAD
INVESTIGATION RESULTS
105
E 00
E 00
eo, 2/903S
6.00
6.00
----
V.. FLUTTER PRESENT
EXPERIMENT
o 5 M/S
A 11.5 M/S
4
MOAMONv
4.00
Q 16 M/S
T
4. 00
e
T
WA.
A/W
E
000
A6
C
S
0
0.40
G
0
0
0.80
1.28
1.68
ROOT AM (DEC)
Figure C.1
A Avv.e
10
0
0
0
0
.0
2.09.
E aI
[02 /90]s layup wing, steady airload deflection.
0 0 0
0
0.48
0.60
1.28
ROOT ABA (DEC)
1.60
2.0
E 81
E 00
6.00
E 00
2.00
E+15 ,2/8S
---- CW
VvV FLUTTER PRESENT
EXPERIMENT
0 5 M/S
A 11.5 1/S
O 16 f/S
-A4
0.00
T
N
I
S
T
4.00
A -2.00
H
G
0
E
G
2.00
-4.00
000
0.
.00
~6.001L-4
0.40
0.80
1.20
RfT AA (OFG)
Figure C.2
1.680
2.88
F Al
[+152/0]s layup wing, steady airload deflection.
0.08
0.40
0.80
1.20
RlT AMA (MfG)
1.680
2.00
E At
E OO
6.08
E 00
2.00
C+-15/03S
FLUTTER PRESENT
vW
1.00
EXPERIMENT
0 5 /S
A 11.5 IS
o 16 W/S
0.00
T
4.00
I
A
I-1.00
S
T
A-2.00
N
G
D-3.00
E
G
A
2.00
A
Aa
-4.00
-5.00
8.40
8.88
1.20
RMT AA (DEG)
1.68
2.88
-6.00
8.00
E aI
Figure C.3 [±15/0]s layup wing, steady airload deflection.
I
i
a I ,
i II •I ,a a
1 I
0.80
8.48
I
a
a
a
a
I i
I I
1.20
RMT AnA (CDEG )
i
a
a
5
I
I a1 i
1.68
s
1
1
II
I
2.00
E at
E 0
6.00
E 0
6.0
E-+15/03S
----MAL
VWv
FLUTTER PRESENT
EXPER I ENT
0 5 IM/S
A 11.5
S
Q 16 M/S
W
I
4.00
S
vvONw
T
A
H 2.00
G
D
E
G
2.00
0
0
-2.00
0.40
0.80
1.28
RMT AOA (DEC)
Figure C.4
"
8.08
1.68
E l1
[T15/0]s layup wing, steady airload deflection.
0.40
8.80
1.20
RO T AM (EG)
1.68
2.0088
E 81
E 01
E 0-
,,NV
6.00
FLUTTER PRESENT
1.2g
E-15,2/3S
i
I-I
M/S
00A 516
M/S
11.51,M/S
T 0.80
S
%MAW
G
C
M
0AAAV
0.40
0
E
2.00
A
G
0EG
to
.00
0B.0
1.40
0.80
oIa Q
1.20
ROOT AOA (DEC)
Figure C.5
1.60
00.00
2.00
E at
O
O
A
"''0.00
[-15 2 /0]s layup wing, steady airload deflection.
0.408
.88
ROOT AD
1.20
(DE)
1.68
2.00
E
81
E 01
E 01
E+38,2/E3S
e.e00
EXPERIEHT
0 5 M/S
A 11.5 Mt/S
1.20
0
6 16 M/S
T
H - ACa
W
\
0
0
A
A
0
0
0
-,
0.8A
A
C
11
G
0D
0
E
4-8.88
0.40
a
A
A
S
0.40
l
0.80
1.20
ROoT AA (EC)
Figure C.6
1.60
2.00
E
8.0088
at
[+30 2 /0]s layup wing, steady airload deflection.
8.40
8.80
1.28
ROOT AA (DEC)
1.68
2.00
E at
E 01
E0
2.00
---- C
IAL
\VVV
FLUTTER PRESENT
EXPERII'ENT
0 3 riS
1.28
A 11.5 M/S
o 16 M/S
0.00
T
w
\m
W 0.80
4
I
S
T
A -2.00
C
M
D
E
G
0.40
A
0.00
A
a
J
-4.00
'Pa
0.00
0.40
0.80
ROOT A
Figure C.7
1.20
(DEG)
1.68
2.00
E 8I
[±30/0]s layup wing, steady airload deflection.
0.40
0.80
1.20
ROOT AA (DEG))
1.60
2.00
E al
E 00
6.00
E 01
E-+30/03S
\AvV,
1.20
FLUTTER PRESENT
EXPERIMENT
o 5 IS
A 11.5 M/S
o 16 M/S
VwVONV
0.80
vw*O^"
0.40
.00
0.00
e.40
0.80
ROOT
Figure C.8
1.28
IO (DEG)
1.60
2.00
-2. 00 L0.00
E 81
FF30/0]s layup wing, steady airload deflection.
0.40
0.8e
1.20
RFT A0A (DEC)
1.68
2.00
E al
E 01
1.20
E 01
E-30 2/81S
---- **LY
FLUTTER PRESENT
WNVA
EXPERIMENT
0 5 M/S
1.20
A 11.5 M/S
0.80
wA W
0.80
8.40
0.40
0.00
0
.0-
0.48
0.80
0
1.20
RMT AA (DEG)
Figure C.9
0
0
1.68
0
2.00
E
0.00
I1
[-302/0]s layup wing, steady airload deflection.
0.48
0.80
1.20
RfOT AM (MEFC)
1.68
2.008
E a1
E 01
E 01
E+45, 2/03S
--.-- H
\,V VV FLUTTER PRESENT
EXPERIMENTI
M/S
0.00
0
A,
0l
0
1.20
M/S
o
11.5
A 16
I
4
0
0
0
0
,
,
0
0
a
T
I
W
0
G
0.40-
0.000.00
Figure C.10
G -0.80
0.40
0.40
0.80
ROOT A
-,.
1.20
(DEG)
1.60
2.00
E al
,, , ,
sI
.00
[+452/0]s layup wing, steady airload deflection.
0.40
,,i,
0.80
1.20
RKOT aA (DEG)
,,,,
1.60
2.00
E Iat
E O1
E 00
2.00
.
E+-45/3S
WP/''t/
1.20
-
FLUTTER PRESENT
E)PERIMENT
o 5 M/S
A 11.5 MS
O
0 16 M/S
O
O
0
4
0.
T
N
m
A
0.4oG
e -
E
G
A
A
-4.00
I
-6-
e.00
0.40
0.80
1.20
ROOT AOA (DE
Figure C.11
)
1.60
2.00
0.00
E BI
[±45/0]s layup wing, steady airload deflection.
0.40
I
0.80
I
1.20
ROOT AM (CDEG)
1.60
2.00
E aB
E 00
6.00
E 01
C-+45"83S
--- AtAL
~W' FLUTTER PRESENT
EXPERIMEHNT
1.20 -
O 5 M/S
A 11.5 M/S
O 16 M/S
4.80
v0 O ' /
T
W
I
S
T
W 0.80
C
A 2.00
14
G
M
A
0.40
0.800 -
0.00
0.88
0.40
0.80
1.2
ROOT AOA (EGC)
Figure C.12
D
E
G
1.60
2.00
E 1I
[F45/0]s layup wing, steady airload deflection.
0.40
0.80
1.20
ROOT ADA (OFG)
1.60
2.80
E 81
E 01
1.20
E 01
L-45
1 2/3-
1.20a
11.
M/S
a
T 0.80-
T
C
M
1
aG
0.40
E
G
A
0.40
0
0
0.00
0.40
0.80
ROOT A
Figure C.13
1.20
(DEC)
1.60
2.00
E
0.00
I1
[-45 2 /0]s layup wing, steady airload deflection.
0.40
0.80
ROOT
1.20
M (Ef)
1.60
2.00
E
1t
APPENDIX D
FLUTTER
INVESTIGATION RESULTS
119
Table D.1
FLUTTER DATA
WING
UNSWEPT WING (A = 0) (w and 0 are at the wing tip)
V
(m/s)
[02/90] s
[+152/0]s
[+ 15/0]s
[U15/01]s
Aw
0
AG
(cm)
(cm)
(deg)
(deg)
10.3
a
r
w
(deg)
avg
avg
26
1
0.5
0.7
1.0
13
10
2.4
0
3.5
9.8
25
1
0.2
0.2
0.7
8.1
24
10
5.1
0.2
-4.0
28
3
4.0
0.8
-0.7
8.5
16
10
3.8
0
0.4
5.0
4.3
0.5
4.8
14
2.8
7.0
21
10
10
2.2
0.1
18
1
4.7
9.4
13
10
3.3
29
0
-0.6
1.7
2.4
28
12
7.9
4.0
-4.8
[±30/0]s
27
10
11.7
1.2
-3.7
[F30/0]s
24
1
13.5
1.3
.13.5
15
10
8.4
0.2
[-152/0]s
[+302/0]s
11.7
120
10.1
7.9
5.6
w
(hz)
19.3
17.4
5.8
28.4
7.6
8.8
6.7
Table D.1
(continued)
UNSWEPT WING (A = 0) (w and 0 are at the wing tip)
FLUTTER DATA
WING
V
(m/s)
[-302/0]s
[+45 2 /01s
[±45/0]s
[F45/0]s
[-452/0 ]s
a
r
w
avg
Aw
0
(deg)
(cm)
(cm)
(deg)
AO
w
avg
(deg)
(hz)
2.6
15
1
5.1
17.2
3.9
8
10
3.0
4.5
4.9
27
1
2.1
3.6
2.2
23
10
9.9
2.7
-2.7
25
8
13.1
0.4
-2.4
3.2
44
18
10
10.8
0.6
-4.2
6.8
49
22
1
16.0
0.9
9.0
1.5
40
14
10
12.2
0.1
6.3
6.3
55
14
1
10.9
9
10
5.2
12
10.3
121
10
5.2
4.9
5.8
20
14
9.1
26
-
3.5
4.2
Table D.2
FORWARD SWEPT WING (A = -30) (w and 0 are at the wing tip)
FLUTTER DATA
v
WING
(m/s)
a
r
(deg)
w
avg
(cm)
Aw
(cm)
avg
(deg)
AO
(deg)
(hz)
20
1
8.8
0
6.1
1.5
18
10
8.3
11.0
1.5
3.0
21
10
25
1
18
10
19
1
6.2
11.9
4.8
9.5
4.7
15
10
5.2
12.8
4.7
9.3
7.5
15
1
3.2
15
8.3
23.4
4.0
12
10
3.9
12.9
7.7
18.1
6.8
[+30 2 /0]s
21
10
14.5
0.8
-18
12
59
[+30/0]1s
22
1
19.8
0.7
-6
44
33
20
10
21.5
1
-7
42
31
17
1
6.9
19.7
4.0
12
3
13
10
7.5
9.5
2.5
5
[02/90] s.
[+15
2
/0]s
[±15/0]s
[15/0]s
[-152/0] s
[ 30/0] s
12.2
1.1
-10.7
16.7
40
7.8
0
-1.6
27
44
8
0
-0.4
14.3
50
122
5.6
Table D.2 (continued)
FLUTTER DATA
WING
FORWARD SWEPT WING (A = -30)
V
a
(m/s)
[-30/0]s
r
(deg)
11
1
8
10
[+45 2 /0]s
[±45/01s
[F45/0]s
[-4520]s
w
avg
(cm)
Aw
(cm)
(w and 0 are at the wing tip)
avg
(deg)
4.1.
18.1
7.0
3.7
10.9
3.5
16.9
0.2
6.9
14.5
AO
(deg)
22
(hz)
2.7
4.6
15
10
16.9
0.7
20
1
19.5
3.0
-10.5
5.7
14
10
12.9
6.3
3.5
4.8
14
2
9.5
19
5.5
2.8
11
10
9
14
1.5
4.3
10
1
7.1
8.6
8.4
8
10
11.6
7.5
6.9
123
19.1
14.7
3.5
30
- 60hz
A = 0
DEGREES
33
++302/0]s
32v
38
20
5.9
52
1+152/0]s
[+452/0]s
>4
F-A-2
o
2.6
7.6
3.5
[-152/0]s
4.1
10
4.9
ANC;I.E
Figure D.1
A=0,
OF
[02/90]s
12
8
4
0
39
[-452/0]s
[-302/s
ATTACK
(DEC)
crossection of test wings, flutter boundary curves.
16
20
20
[+152/0]s
448
5.9
7.6
53
[-152 /0]s
-
1±15/0]s
10
1IT15/0s
AN(GLE
Figure D.2
A=0,
12
8
4
0
OF ATTACK
16
(DEC)
15 degree ply fiber angle layup wings, flutter boundary curves.
20
30
60
A = 0 DEGREES
60
40
[+302/s01
[4730/0]s
20
H
H
7.2
2.6
54 130/O]s
547.5
10
4.9
[-3 0 /0]s
2
8
4
0
ANCIE
Figure D.3
A=O,
OF ATTACK
12
10
(DEC)
30 degree ply fiber angle layup wings, flutter boundary curves.
20
A = 0 DEGREES
52
20
[+452 /Ols
3.5
[±45/0]s
-4.1
10
[-452/0]s
AN
Figure D.4
A=0,
12
8
4
0
CI.E
OF A'I'TTACK
16
(DEC)
45 degree ply fiber angle layup wings, flutter boundary curves.
20
59hz
A = -30 DEGREES
[+ls,/O]s
20
4.0
U
[+302/ O]s
59
42
44
[+452/0]s
00
2.7
8
6.8
0
ls
[-1520]s
[02/90]s
2.0
3.5
4.6
0
4
8
AN CI.E
o1I ATTACK
12
[-45,/0ls
[-302/01s
lh
(DEC)
Figure D.5 A = -30, crossection of test wings, flutter boundary curves.
20
A = -30 DEGREES
35hz
43
4.7
[+152/0]s
4.0
53
[115/0 s
7.5
6.8
[-152/O]s
44
[TI5/0]s
ANCLE
Figure D.6
A
OF ATTACK
(DEC)
= -30, 15 degree ply fiber angle layup wings, flutter boundary curves.
A
= -30 DEGREES
32
33
[+30 2 /0]s
3
6.4
Lt30/0ls
~
--
2.7
S;30/01s
4.6
[-302/0]s
I
I
AN:IE
Figure D.7
I
.
F ATTA'I"I'CK
(DEC:)
A= -30, 30 degree ply fiber angle layup wings, flutter boundary curves.
A
= -30 DEGREES
40
5.7
[+452/0]s
2.5
2.0
4.7
-4.3j4
0
[T45/O]s
[-452/Ojs
___
AN;:IE OF ATTACK
Figure D.8
__
(DEc;)
A= -30, 45 degree ply fiber angle layup wings, flutter boundary curves.
80
r-
2B
1T
IB
0
0
20
40
VELOCITY
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D. 9
A= 0O [02/90]s layup wing, V-g diagram.
132
60
N
40
z
w
20
0
20
40
VELOCITY
( M/S
)
0.4
0.2
0
-0.2
-0.4
-0.6
Figure D.10
A=
O, [+15 2 /0]s
layup wing, V-g diagram.
133
40
20
VELOCITY
( M/S )
0.4
0.2
0
-0.2
-0.4
-0.6
Figure D.11
A= 0, [±15/0]s layup wing, V-g diagram.
134
80
2B
60
N
-
40
B1T
z
20
0
20
40
VELOCITY
0.4
-
0.2
-
( M/S
60
)
1T
0o
00
20
40
z
-0.2
IB
=
-0.4
-0.6
Figure D.12
A=
0, [:15/0]s layup wing, V-g diagram.
135
60
80
60
2B
1T
N
40
-,
z
o
20
1B
0
20
440
VELOCITY
60
( M/S )
0.4
0.2.
IT
0
U-
0
S40
60
Z
-0.2
2B
SI
1B
-0.4
-0.6
Figure D.13
A=
0, [-15 2/0]s
layup wing, V-g diagram.
136
80
r-
I1T
2B
1B
SI
I
I
VELOCITY
( M/S
I
)
0.2
0
-0.2
-0.4
-0.6
Figure D.14
A=
0, [+302/0]s layup wing, V-g diagram.
137
I
80
60
40
20
00
20
0
40
VELOCITY
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D.15
A=
0, [-30/0]s layup wing, V-g diagram.
138
60
1T
z
=
c,
20
1B
0
0
20
40
VELOCITY
0.4
60
( M/S )
IT
0.2
0
0
S20
60
z
A
-0.2
2B
3
-0.4
1B
-0.6
Figure D.16
= 0, [F30/0]s layup wing, V-g diagram.
139
80
60
1T
2B
N
40
Z
m
20
IB
0
0
I
0
I-
I
20
40
VELOCITY
60
( M/S )
0.4
0.2
2B
2B
1B
-0.4
-0.6
-
Figure D.17
A=
0, [-302/0]s layup wing, V-g diagram.
140
40
T
2B
20
lB
I
0
I
I
VELOCITY
( M/S
I
)
0.4
0.2
0
-0.2
-0.4
-0.6
Figure D.18
A=
0,
[+452/0]s layup wing, V-g diagram.
141
I
2B
1T
1B
VELOCITY
( M/S
)
0
-0.2
-0.4
-0.6
Figure D.19
A=
0, [±45/0]s layup wing, V-g diagram.
142
60
N
40
20
0
20
40
VELOCITY
( M/S
)
0.4
0.2
0
40
-0.2
2B
-0.4
-0.6
Figure D.20
A=
0, [:45/0]s layup wing, V-g diagram.
143
60
40
20
0
20
40
VELOCITY
( M/S )
0.4
0.2
0
-0.2
-0.4
-0.6
Figure D.21
A= 0, [-45 2 /0]s layup wing, V-g diagram.
144
2B
IB
I
I
I
VELOCITY
I
I
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D.22
A= -30, [02 /90]s layup wing, V-g diagram.
145
I
r
z
a
20
20
40
VELOCITY
( M/S )
0
-0.2
-0.4
-0.6
Figure D.23
A=
-30, [+152/0]s layup wing, V-g diagram.
146
80
60
N
2B
40
z
3
1T
20
1B
0
0
20
40
VELOCITY
0.4
60
( M/S )
-
0.2
S1IT
00
O
60
40
1
-
2B
-0.2
3
-0.4
IB
-0.6
Figure D.24
A
=
-30,
[±15/0]s layup wing, V-g diagram.
147
2B
1T
1B
VELOCITY
( M/S )
0.2
0
" I
20
I
I
i
40
6C
2B
-0.2
1B
-0.4
-0.6
Figure D.25
A=
-30, [f15/0]s layup wing, V-g diagram.
148
1T
IB
"
--------VELOCITY
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D.26
A=
-30,
[-152/0s layup wing, V-g diagram.
149
z
20
00
20
40
VELOCITY
( M/S )
0.4
0.2
0
-0.2
-0.4
-0.6
1
Figure D.27
A=
-30, [+302/0]s layup wing, V-g diagram.
150
20
0
0
20
40
VELOCITY
( M/S )
0.4
0.2
00
-0
-0.2
B
S-0.4
-0.6
Figure D.28
A=
-30, [±30/0]s layup wing, V-g diagram.
151
80
1T
60
2B
_
z
40
20-
20
0
20
40
VELOCITY
60
( M/S )
0.4
0.2
0
20
40
60
IT
z
-0.2
2B
T
1B
-0.4
-0.6
,Figure D.29
=
-30, [F30/0]s layup wing, V-g diagram.
152
20
40
VELOCITY
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D.30
A= -30,
[-30 2 /0]s layup wing, V-g diagram.
153
z
20
0
20
40
VELOCITY
( MIS
)
0.4
0.2
S
0
-0.2
S-0.4
-0.6
Figure D.31
fA= -30, [+45 2 /0]s layup wing, V-g diagram.
154
=
o"
04
20
0
rL
20
40
VELOCITY
( M/S
)
0.2
0
-0.2
-0.4
-0.6
Figure D.32
A=
-30, [±45/0]s layup wing, V-g diagram.
155
IT
40
2B
20
IB
0,
VELOCITY
( M/S
)
U0.
0.2
-
I
S,
0
20406
1T
-0.2
2B
lB
-0.4
*-
-0.6
Figure D.33
A=
-30,
['-45/0]s layup wing, V-g diagram.
156
40
20
00 0o
20
40
VELOCITY
( M/S )
0.2
0
-0.2
-0.4
-0.6
Figure D.34
A=
-30, [-452/0]s layup wing, V-g diagram.
157
ABOUT THE AUTHOR
Captain
Brian
an
is
Landsberger
J.
Air
fighter
Force
pilot.
in 1972, up
Following graduation from the United States Air Force Academy
is
to April of 1981 he has been flying for the Air Force. His expertise
OV-10 Bronco.
in tactical aviation, having flown the F-4 Phantom and the
he was a
This includes two assignments to The Republic of Korea where
forward
controller,
air
Infantry
Divison and a
Squadron.
Advances
applications
went
to work
from
in
Korea, he
the
the
for
officer
26th
Korean
497th Tactical Fighter
entered
the
Massachusettes
under the sponsorship of the Air Force Institute
While at M.I.T. he has worked in the M.I.T. Laboratory
aeroelastic
on
concentrating
(TELAC),
Composites
of Technology.
for
liason
Flight Commander
After returning
Institute of Technology,
air
an
for
for
composite
the
From M.I.T.
material.
Flight
Test
Center
California.
158
Captain
at Edwards
Landsberger
Air Force
Base,