e. ,. ,,. ,. ,. Indicial lift due to impulsive plunging, circular planform,. M = 0 . ..... [7] , later showedthe relationship betweenWagner's solution and Theo- solution [8] for the ...... Aerodynamics for Aeroelastic. Analysis of Interfering. Surface_. AGARD.
NASA
NASA
CONTRACTOR
CR-2779
REPORT O_ r,,,. r_ l
Z
FINITE OF
STATE
AEROELASTIC
MODELING SYSTEMS
Ra.ja. W?a Prepared
by
STANFORD Stanford, for
Langley
UNIVERSITY Calif.
94305
Research
Center
NATIONAL AERONAUTICSAND SPACE ADMINISTRATION •
WASHINGTON, D. C. • FEBRUARY 1977
1. Report
No.
NASA 4. Title
2. Government
Accession
No.
3, Recipient's Catalog
and
Subtitle
FINITE
5.
STATE
MODELING
OF
AEROELASTIC
SYSTEMS
Authorls)
Performing
Organization
S_nsoring
Name
Agency
Supplementary
Name
and
10.
Work
'11.
Contract
94305
Code
Organtzation
Unit
Report
No,
No,
or Grant
NGL
No.
05-020-243
13.
Type
14,
Sponsoring
Address
of
Report
and
Period
Contractor
Aeronautics DO 20546
and
Space
Administration
Covered
Report
Agency
Code
Notes
Adapted from Ph.D. Dissertation, May 1975 Langley technical monitor: Robert V. Doggett, Jr. i 16.
1977
Organization
and Address
University California
National Washington, 15.
Performing
Vepa
Stanford Stanford, 12.
Date
8. Performing
Ran jan 9,
RePort
February 6.
7.
No,
CR-2779
Topical report.
Abstract
A general theory of finite state modeling of aerodynamic loads on thin airfoils and lifting surfaces performing completely arbitrary, small, time-dependent motions in an airstream is systematically developed and presented. In particular, the nature of the behavior of the unsteady airloads in the frequency domain is explained. This scheme employs as raw materials any of the unsteady linearized theories that have been mechanized for simple harmonic oscillations. Each desired aerodynamic transfer function is approximated by means of an is, a rational function of finite transform variable.
appropriate degree
Pad_ approximant, polynomials in the
The modeling technique is applied to several and three-dimensional airfoils. Circular, elliptic, and tapered planforms are considered as examples. functions are also obtained for control surfaces three-dimensional airfoils,
117. Key
Words
(Suggested
by Authorls))
18.
Unsteady aerodynamics Aeroelasticity Active controls
Distribution
two-dimensional rectangular Identical for twoand
Statement
Unclass
i f i ed-Unl
im i ted
Subject 19.
_urity
Classif.
Unclassified
(of this re_rtl
20.
Security
Classif.
Unclass
(of
this
page)
i f ied
that Laplace
21,
No.
of
Pages
188
22.
Cateqory 39
Price"
$7.00
* For sale by the National Technical InformationService, Springfield, Virginia 22161
INTRODUCTI
ON
General
............................................. l)c's_-r-il_t
Rev icw _) ,qllnl!)lnF,/
io_.
,_!
Pc_-ctincnt i
i[
(kn_,t C[_lt[
1
!i_.
l'r_,t)tem
_:]',][ill
i_.i S
t
2
..................
3
. .....................
............................
5
SYNBOIff-] ................................................ FINLTE
SfA'[F,
W)l)l'!,!h',;
Finite
FINITE
S'I'AI'I] '['wo
c_i
} 0
This compute is
equation
K(t)
the
use
for
suggests
T _
of discrete
forms
[31].
brief
discussion
of
To
the
domain, at
the
In
finite
2T
for
Fourier
order
observe at
t _
that
to
the
but
the
method
f
N = TF, the
each
An
like
equivalent
used
to
possibility
the
fast
Fourier
of
this
technique
disadvantages
sampling
both
in
the
above
frequency
intervals,
the
j.At,
j = O, +_ i,
+_ 2,
..., +_ (N-I)
= n'Af,
n =
+ 2,
..., +
O, + i,
n
for
T.
be
transa
equation
and
is
time
evaluated
points,
tj =
where
given
may
is presented.
of
equal
any
quadrature
transforms
show
effect
numerical
spectrum
with
highest
cycle.
_
F = 1/At
frequency
With
Cf(f)
the
are
and
assumption
periodic
with
to be that
At. is chosen J sampled
the
period
T we
Wg(j.At)
N-I )e+2Wi 1 _ Cf(n. At : _ n=O
Cf(n'Af)
T = _
N-I E W_(j.At)e n=O _
A
We
then
n
A Cf(n,Af),
have
the
Xj
transform
= N-I E Awn_+n-J {xj ] = n=O
N-I An=
I Z X.w, N j= O 0
small
least function
twice and
enough during the
have,
n.j/N
-27ri n.j/N
2_i/N
Xj A TWg(j ,At) and w
discrete
at
indicial
-
Let
(N-l)
-
T = i/Af.
present
and
e
pair,
[fl]{An }
-nj {A n ] =
[_]-I{xj}
21
From the orthogonality function it
properties
and the properties
of the exponential
is possible to factor
[_] in such a manner that the number N of computations involved for inversions are reduced from N2 to _ log2N _rovided N = 2m). This decomposition technique is the Fast Fourier transform.
Thus it is possible to obtain numerical values for Wg(tj) for
various values of tj by using different degree of accuracy.
sampling periods T to any desired
Also errors due to the assumption of periodicity
can be corrected for_using WindowTechniques. Howeverthis does not lead to a finite indicial
state model. From the numerical data generated for the
functions_ Kl(t),
it is essential
number of exponentials (2 or 3). [32].
to fit
the data with a finite
This is usually a non-linear
technique
The Laplace transform of such an approximation would then lead to
a finite
state model in the frequency domain. This whole procedure is
undoubtedly extremely tedious and involves several computational steps. Conceptually the procedure involves approximating a distributed
parameter
system by a discrete time system which is then approximated again by a continuous system in the time domain. Thus a direct procedure of finite state modelling from knownvalues of the response at f = fn' with the correct initial
and steady state behavior is desirable. In order to develop
such a technique_ Theodorsen's function is analyzed in detail. Analytic continuation will
of Theo_orsen's function:
nowbe evaluated in the left
axis and the physical significance this case, _(s) may be written
22
as
The function _(s)
half of the 's' plane along the real of the results will
be explained.
In
¢(s)= 1 _e-S 0
I+Lt
cosh
cosh
v
e-S cosh
v
dv
v dv
0
The
integrands
domain
of
of
v.
evaluated
by
the
two
Further using
they
result
is
of
t = O,
the
wing
motion
with
no
possible. plane
the
case
for
the
pressure
axis
of
of
Asymptotic Krummer
for
and
on
from
distribution
an unperturbed
negative
is the
As the
Clearly,
real _(s)
wing,
exact
axis = l,
the
solution
may
that
at
time
converging
the
is the
is not 's'
solution
quasi-steady on
be
.
state of
the
to
: 0
assumed
in
limit
leads
Im(s)
unperturbed.
of
the
immediately
It was
unperturbed.
increasing
Hence
(s) < O,
the
translation
the
solution
negative
's' plane.
expansions
functions
Re
which
importance.
that
steady
monotonically
divergent.
components
remains
the
both
stationary
implies
'motion'
for
real
: 1
oscillatory
are
rule_
fundamental
was
This
the
are
L'Hospitals
#(s)
This
integrals
we
of Wagner's
indicial
function:
In
terms
of
have 1
_(s)
=
Re(s)
i+½ The metric defined
Kummer
functions for
functions that
-v < arg(z)
l, 2s) s (1½,3, 2s) U(a,
are
> O
n,
multiple
< v.
The
z) are
a class
valued order
with of
the
of
the
confluent principal
singularities
hypergeobranch for
large
23
and small values of z are determined by a and n respectively. As a consequenceof the properties
of Kummerfunctions $(s) maybe
represented as an asymptotic series for large s as_ N+4
: 1 - (4sU(1,1,4s) + E &.
Bi (Cis)U
(]. ,1 ,Cis
1 0 (s)
)+
i=l
N : 0,
where
B i and
C i are
s the
expression
constants.
also
Apart
satisfies
T,t
the
from
being
l,
asymptotic
2,
...,
for
large
condition,
: l
s-*0
Further_ easily for
P(T)
first
Garrick. written
inverse
found.
Wagner's
where
The
the
Hence
Laplace it
indicial
terms
In general
is also
to
K(t)
2 T + I_ +
are the
=
1
+ 1.28435T
2 + 1.8428_
identical
to the
correction
to
asymptotic
find
In particular_
2
rational
[N + 4, N]
24
possible
the
an for
series
asympZotic
can
be
expression
N = 0,
3
T 768 'P(_)
3 +
4.09134T
indicial
Garrick's
function
4
given
approximation
by
may
be
as,
K(t)
The
of
function.
= i + 0.875T
two
transform
-
1 + _
function Pad_
3 = _
in the
approximant.
(Polynomial (Polynomial
brackets Baker
of of
degree degree
is usually [33] has
N in r) N+4 in _)
referred
discussed
the
to
as
a
methods
of
construction
and the properties
of Pad_ approximants.
Pad@approximants of Theodorsen's function: W. P. Jones [34] gives the following
series for _(s): 2
_f(s) z i + ys + y2s + s where y = log(s/2)
+ _.
For small values of s,
The correct asymptotic series for small values
of s is given by,
_(s)
= 1 + ys + y2s2
2 Y 2
+ s3[ z_l
+ sS(y 5-_y3
+
4
+_ _
y3
+ y3+_ ]+ s40
Obviously this
it
series.
is quite
difficult
to
for
large
values
1 + 8s 2
7 64s 3
19 1282
However
construct of
a Pad_
approximant
s, we have
the
from
asymptotic
expansion,
l(i + i _(s) : _
143 + 512s 5
629
8273081
i024s _
+
4194304s
Re
The
fact
s = 0,
that
suggests
the
lag
that
function it may
be
has
a value
approximated
of by
0.5 an
for
s = _
[N_N]
(S)
and
>
+ o(-_)) 7
0
i for
Pad_
25
S
approximant quite accurately.
To demonstrate this,
the first
four
Pade approximants are constructed from the asymtotic series for large s.
They are asymptotic to the lag function uptothe first, third, i fifth and seventh orders in --S respectively. They are given by s+o.5 2s ÷
s2 + 1.5 s ÷ 0.375
0.5
'
2s2 + 2.5s + 0.375
s3 ÷ 3.5 s2 ÷
2.7125s
2s 3 + 6.5s 2 ÷
s
4
4
÷
4.25 s
4.64696s
3 +
9.33371s
8.79392s
3 +
16.71894s
0.46875
÷
0.46875
2 +
5.51735
and
s + 0.49334
, Re(s) > 0 , 2s 4 ÷
These
expressions
half
of
good
for
series
the
This
large
s.
method
of
Wagner
indicial
the
method
suggested
in the
theory
shells
in
method
axis least
N(s)
D(s)
may
26
be
function by
the
of
s =
determined
the they
rapidly
above are
obtained
from
Theodoresen's
[35]
unsteady
over
for
the
approximations
approximately
Luke
÷ 0.49334
is
function
evaluating
aerodyn_nics
are
the
similar
of
entire
right
not
very
asymptotic
and
evaluating
in principle
the Randall oscillating
to
function cylindrical
flow. construction the ik,
squares are
s as
7.67296s
converge
approximating
Since
suitable and
of
supersonic
generalized. imaginary
for
to
However
values
the
The
found
's' plane.
small
for
are
2 ÷
lag the
function Pad_
technique
polynomials by
of the
minimizing
in
Pad_
approximants
is known
exactly
approximant . s,
If C(k) the
may
be
may along
constructed
= N(ik)(D(ik))
coefficients
of
be the by
a
-1 , where tne
polynomials
m z (D(ikj)C( j)-
2
j=l
where
kj,
This
j = i,
technique
points
in
cases,
where
This linear This
only
the
aerodynamic
may
then
for
the
results
be
the
values
values
domain_
the
procedure
Clearly
..., m are
frequency
system
"
3,
needs
the
problem
2,
of
and
loads
reduced
hence can
to
coefficients
C(k)
be
the of
may
of k at
at a finite be
points.
number
generalized
calculated
solution
the
n different
of
to
other
numerically
of
an
polynomials
over N(s)
only.
determined and
D(s).
in
s4÷
0.761036s3+
0.i02058s2+
0.00255067s
+
9.55732
× 10 -6
2s4÷
1.063996s3+
0.i13928s2+
0.00261680s
+
9.55732
× 10 -6
Wagner's
indicial
function
may
be
approximated
Re(S)
> O.
using
the
formula
4 K(t)
where
_i
are
the
roots
: Z -I _(s)__= s
of
the
1
-
denominator
Ai
_ i=l
-BiT A.e 1
polynomial
where
_i
0.011285763
0.0044482234
0.043280564
0.027697193
0.21639860
0.096054968
O.229O35O8
0.40379780
27
By taking the Laplace transform of the approximation for Wagner's function obtained by R. T. Jones [i0],
@(s) =
The
[4,4]
Pad_
indicial
function
results
of R.
Jones
domain. R.
T.
it
can
Jones be
It mations that
the
T.
simplicity is
quite
cut
a continuous
Application theory
to
gust
to
It must
be
emphasized
portion
of
the
a sequence sodially
of
[N,N]
convected Sears
that
Pad@
exact
values
compared
to
the
the of
negative lag
of
the
values.
a downwash
the
one
this
and
the
and
the
given
by
method
is
six
This
input
is
real
axis
function
response
only
problems:
time
that
approxiindicates
is a
it represents
In applying
a
certain
amount
the
Laplace
transform
pressure
distribution
approximants.
This
does
of
can not
a gust
velocity
of the
the
caution of be
the
form,
above is essential.
temporal
approximated
hold
gusts.
[9] considers
corres-
surfaces.
poles
real
the
for
to
lifting
negative
since
frequency
advantage
that
response
problems,
circulatory
the
of poles.
gust
response
are
expression
distribution
and
in the
dimensional
Further
exact
squares
the
both
main
to note
pressure
the
The
function
stable.
of
this
three
lag
to
> 0
4 - 8).
method
to
Re(s) _
least
compared
of
for
transient
by
good
interesting
the
0.0273 0.0279
is very
is also
asymptotically branch
are
obvious.
generalized
+ +
obtained
(Figures
approximation The
0.5615s 0.6910s
approximant
ponding
This
28
_+ 2s2+
we have
for
sinu-
by
WG = Woeie(t-x/V) and showsthat lift
to be equal to Wo ic_c
L(k)
where
S(k)
Kussner's
=
indicial
Wagner's
indicial
exponential a
(Jo(k)-
squares
Laplace indicial
4_ q'b.
function
function
is
function
of Pad6
we
related
to to
Kussner's
the
S(k)e -ik c(k).
function
approximants
technique
obtain
S(k)
iJl(k ).
is related of
smoothing
transform
-_- e
iJl(k))C(k)+
approximation
[N,N-I ] sequence
least
=
of
However one
above
and
expression
same
to
should
S(k)e -ik
developed following
in the
In
obtain
an
fact,
using
the
K 2(t)
= 1
-
Kussner's
-_.T 1
Z Aie i=l
where Ai
Bi
0.012994467
0.0049896174
0.062319920
0.0349314o0
0.40920539
0.13796713
0.51548O22
1.1645811
the
inverse
K2(t ), 4
as
construct
taking for
manner
29
In order to showthe relationship
between the temporal and spatial
portions of the gust response, it is essential to consider a gust of for_n_
i_t WG Kemp
[36]
shows
the
lift
- iFx/V
= WOe to be
of
the
form
W
Vo b
:
where
K_,k)
=
and
k = _-
.
The the to
((Jo(k)-
first
second
is
term the
approximate
function
in
iJl(k))C(_+
the
virtual
C(k)
by
corresponding
H(t) A
steady produce
is
an
[N,N]
to
a downwash
of
a lift
convected
attack equal
G, of to
L(k)
where
3O
T_)
K(k_X)
Pad_
is the Clearly
approximant of
the
circulatory it
is
part
now
to obtain
while
sufficient
the
indicial
form
H(t)
a step'function.
sinusoidally angle
Jl(k))
effect.
W G -- WO e'iFx/V
where
_
function inertia
ei_t
= C_)(J0
chordwise the
form,
gust
past
U G = UO elf(t-x/v),
[37],
=
Uo 4_q-_-
_)-iJl(k))+
_be
ic_t T(_
J0(k)+
an airfoil
i2J l(k).
at
a
is known
to
From this result chordinse
gust
we may obtain the indicial
striking
Us
the leading
lift
edge of the aerofoil
at t = 0
X >Vt X x _>
by
of
, _
is
bending,
influence (torsional
the
and
root
Kh
is
coefficients
for
deflection
at
$)
X
cee(x,_)
=
O_
dk i Gj-rO_7+_
dk
, fo_ __>x_>O
i
=o S where
GJ(k)
torsional the
is
the
spring
influence
torsional
constant
coefficients
rigidity
at
the
reduce
distribution
root. to
If
those
the
and
root
given
in
is
K e is rigidly
the fixed_
[1].
65
restraint
APPENDIX B INTEGRAL EQUATIONS FORUNSTEADY AERODYNAMICS
Three Dimensional Lifting
Surface
The source solution of the linearized potential
is given by
47r@=
where the
H(-)
is
tion
of
the
point
velocity the
and
coordinate
coordinate
systems
(_ - v(t 0-
of t),
at
speed
if the
distance
the
the
distance
between
time
measured
from
of
sound.
disturbance
r in
that
(g,_
_ _
r) moving
, along
in
wing_
is
considered.
assumed
to
coincide
the
point
_, C , to).
disturbance
the
of
the
(_,
_,
A unit
moving
acceleration
66
T is
(Xo,Yo,Z0,to)
(_, _, C ) in the
magnitude
T
r is
The
moving
time;
the
the
observawith
a
strength
of
-1/4wr.
are
the
function,
c is the
time
the
(B.I)
point,
and
system
fixed
coordinates
point
is
system v,
step
source
trayerse
coordinate
velocity
the
is affected c can
- r/c) r
disturbance
disturbance A
-H(T
the Heaviside
observation
instant
equation for the velocity
at
disturbance
at
is
(x,
fixed
time
t
to
the
o
will y,
z),
a fixed
with
and
The
fixed
axes
source
produce after
a
moving
= t.
potential
frame
to
x 0 axis
The
acceleration
point,
respect
negative
_, T) referred
reference the
the
with
are at
the
an
some
time
delay;
=
[HC'v'-r/c) i _ ]=-_' l 2
2 r
In fixed tion
=
(x - _ + v(t o-
coordinates,
potential
the
t))2+(y
velocity
+ (z - C),
_ _)2
potential
is
(B.2)
related
to
the
accelera-
as
=
@(x,y,z,t)
_t2_(x,y,z,_)dt tI
where
[tl,t 2]
is the
velocity
potential
last
first
the
and
doublet
at
at
domain
time
instants
observation Therefore
time
of
influence
t in fixed
of
time
at
of
_, which
coordinates,
t 2 and
which
the
moving
affect_ tI are
source
can
the the
affect
point. the
(_, _,
velocity _,
to)
potential
of
strength
at
(x,y,z,t)
A_(_,
_, _,
due to)
to
is
a moving
given
by,
t2 ¢(x,y,z,t)
=
i - I_
_ _
_
A_(_,
_,
_, to)
i _ dt 0
(B.3)
tI
The
time
t 2 satisfies
((x
This integration and
t2
the
- _)
÷ v(t 2-
is the
equation
for
equation
- t = T,
we have
equation,
t))2+(y-_)2+(z-_)2
for
en
(B.3).
ellipse
= c2(t 2-
and
If X = x-
t) 2
determines _, Y
= y
(B.4)
the - _
region
and
Z =
of z -
67
MX+R cT -
where
R =_2
For
+
M < i,
I -M2
(i - M2)(Y 2 +
only
one
of
If the
the
perturbations
the
past
then
unperturbed
tI =
for
_
time
the •
solutions
c
is
negative.
Hence
-R i
- M2
flow
field
existed
for
infinite
the
other
hand
the
flow
0n
t < 0,
_B.5)
(B.6)
t+
in
i
Z2)
IMX t2=
+ R -MX M2 - i
or
then
t2 = tO +
if
time
field
i
MX
-R
c
i
- M2
(B.7) IMX tI =
IMX t < c
i
M2
observation
and
the
period
take
0
provided
t >
c
i
-R -M 2
-R
the
would
is
clearly
and
If
in
'
tl
=
point. of
to pass
t2 = 0, For
influence the
M
as
> i_ both
is the
point
of
time
disturbance solutions an
observation.
IR t2=
the
has for
advancing
not
T are
reached negative
spherical
wave
Therefore,
-MX
t+ c M2
-i
(B. IR+MX tI = t c M2
68
-i
provided,
of
limit), we
we
course,
may
denote
let
both
t I and
M = I +
Eor
U i A= X + V(t i
- t),
X U2 - i
provided
the
For
flow
supersonic
U2
Hence doublet
field
the
at
positive.
- e
i = 0,
i,
and
take
the
2,
then
in
-M 2
' UI
= x
- Vt
for
time
t sufficiently
is equal
at
(x, y,
1
_
moving
velocity
doublets
on
flow,
t < 0.
t)
due
to a moving
U O- X ,t + --V---
)
at
lifting
=
- V1
_/Uoe+y2 + z2
(x, y,z,t) surface
j'j'd_
d_ _z
s The
downwash
on
the
surface
S,
jU2
u1
is _iven
due at
to
a distribution
_ =
0,
A_(_,_,O,t
is
given
4_ w(x,y,t)
d_ S
dy
_ _z
of
by
)
+ T
dUo (B.12)
_/Uo2+y2 + z2
by A¢(_,n,
1Lt _ If = - V z-_0 S--z
(B.II)
dU o
U 0- X 4vr@(x,y,z,t)
If
(B. 9)
J"
potential the
subsonic
z _0.
(B.IO)
z,
A_(_,I]_
u1 total
as
to
-_
The
limit
(transonic
(MR + X) M2 - 1
,UI=
potential
=
M = i,
large,
U2 _(x,y,z,t)
For
-MR
- X1
velocity
(_,_,_,to)
i
is unperturbed
flow,
- MR M2
t 2 are
0,t
+ U -X)dU o
fU2
o
UI /
2
U o
+ y2
+
Z2 (B. 13)
69
It
can
such
be
shown
that
that
a discontinuity
across
a doublet
layer
exists
[I]
A$(x,y,O,t)
=
_(x,y,O,t)
- _(x,y,-0,t)
=
Ap(x,y,t) 0 =J
where
Ap
is
the
pressure
Substituting using
Liebnitz
downwash
in
rule,
subsonic
w(x,y,t) = V
(B.14)
_,n,t
in
(B.13)
have
the
across
and
ff
following
surface. the
integral
differentiation
equation
for
the
flow.
dEdn
[Ap(_,_
S
+--Uo-X i
O
the
performing
M
1 8_q
+ I U Ap
we
differential
(B.14)
V
t + '
I
I(/U
dU
3
2+y2
U_X)
MX+R 1
R I X2+y 2
o
)
O
(B.15)
+ I tAp(_,q,(t-T)) VdT ] _(_-x/v) 0 ((X-VT)2+Y 2) X-MR I where
Ap(_,n,t)
=
0
for
tx T
type
of
airfoil
Ap(_
t
(X+_))
-x
motion.
In
incompressible
d_
(B.26)
v
have
w(x,O) V
Non-dimensionalizing Laplace
> XL
Vt -X
_ 2 d_
qk
time
(l+M) M
if x - Vt (I+M)< M x_
= xT
Eqtmtion
- Vt
transforms
of
M @(x,O)
(B.27)
-
_
q
the
time
variable
the
integral
t to
equations,
T = Vt/b
and
taking
we have
8_(x,s) 1
V
where
#
K2(x
- _,
the
(B.28)
s) _(g,s)d_
q_ w(x,s)
is the
Laplace
transform
of the
downwash
and _(x,s),
77
the
Laplace
transform
of
the
pressure
K2(x,s) = 2s*S£e+S*_[ (_l(Ms* Ixl)+%(M/Ixll
l+_+ 1 s x(x-l) M "- sx _e
+ e-S x_ _n(
K0(Ms * lXlX)dX]]
(B.29),
0
where
s
second
=
s/_ 2,
K0(. ) and
and
K2(X's)
all
Bessel
functions
of
the
has
flow,
a well
distances
known
by
(B.5o)
o0: + 2M_6(x)
26 X
Is- 0 -
incompressible
B.22
modified
kind.
Also K2(X,s)Is-
In
Kl(. ) being
the
(B.31)
equation
inverse
B.29
given
reduces
by,
to equation
(after
B.22.
Equation
non-dimensionalizing
semi-chord).
i
_o(X,S)
(B.32)
= _ a(x,z,_)_(z,s)dz -1
where
G(x,z,s)
= - 2_ [sA(.,z)
1
i -xz+
-x
_
1-_
-_xe./_
i:_
(¢(s)-
- z2
$(s)
78
_ __--_x2
_ xz
Kl(S) : Kl(S)+ K0(s )
,
real
(B.55)
(B.3_)
^(x,z) = _ log i
_ + _---__x )]
2
__z
(s)
>
0
(B.35) continued
2su(1_,3, 2s)
2so(1½, 3, 2s)+
= S "_[i -
[(K0(x)_ KI(X))2+
l, 2s)
2(I0(x)
, real
(s)
continued
+ ii(x))
0
Ki(" ) and li(- ) are modified Kummer
functions
2 -i -xt ] e 2 x
and U(a,b,z)
ax]
is a
function.
Equation method
Bessel
(B.35)
> 0
(B.28) may be reduced
of doing this reduction
equation
may be reduced
with a completely
to several
alternate
was given
by Fettis
to a non-singular
Fredholm
continuous
kernel
function
[70].
forms.
One
The integral
integral
equation
(defined in Appendix
D)
given by,
Ap(x,s)
=
_Po(x,,) ._
KNc(X,_,s)
- s
.M2jf
Kcc(X,_,s)_p(_,s)d_ -i
= H(_ - x)- cos x _ (_(s)-l) 7T IT
(B.56)
l-x
I - I_
_
H(_ - x) is the unit
_-IG(X'Z's)K(s(z-_))dz" step function
and
K20(z- _,s) 2 M2 _(s(z-_) = K2(z - _,s)B + _ Another obtained.
important
representation
The kernel function K2(X,s)
of the integral may be written
equation may be as,
79
:
K 2 (X,s)
M 2 * )-K20 s (X ,s*) _{K2o(X,s*)T[K2o(X,s
e+M2Xs*
+ _-_-.s*.[ --
M2
.2
in
2+B 2 in ___2 _
4
s - 4 *
]
(l-B)in M _ 2¥] M2
2
2 s [_anl+--
_ -
1
-_-_in
-
---
+ o(M4).o(x2+ax log x)} + s *XM 2 = e
* B{K20(X,s
+ M2s* 2
s
where
K20(X,s
M2 )- _[K20(X,s*)-K20S
(l-2 [in (_Ms
*)
= 2(
(B.37)
)+ _])+
½ + s*In
M m V(X,s
(X,s*)]
)]
(B.38)
X)
(B.59)
(B._)
Kv(X,s ) : O(M°+ alnM)
Using
equation
solution
for
B.38
we
may
incompressible
a_(x,s) _Pov(x's*) _s*x q
+
-
V
Bi(s,. x)Ji(s*,x)]-
e
invert
equation
flow.
Thus
we
the
well
known
have
M2 _s*x[ 1 _ A._(s,*x)ji(s,-I) * i:o
M _ 4 eM2s*x
1 _ 1 G(x,z,s . )Kv(Z-g,s J"
)e_M2s*_.
_(_,s)
q
• d_dz
8O
using
- 2"_ e
-i-i
where
B.28
(B. 41)
APov(x's*)
q
i
i
. (B.4_)
= _ j"c(_,_,s)_(_,s)e -M2szdz -I
i Ji (s*'x)
:
7
_M2s* _ _
Ap(_,s) q
X
Ao(S*
x ) =
ln-lx(2s*_s*2x)+
S
_
e
l-_x_x l+x (s*@(s*)in
*
2-(@(s*)-l)
*
+ -_-(_(s
)+s
(l+x))+s*2(l+
in
2)(1
+
x)).
Bo(S ,x) = -(2 - s _)Trs
Al(S
,x) = s*21n-lx
* B l(s
,x) =
=
Thus
+
s*(_(s*)-l)
*2 -TTs
[i - 2(in( 1 Ms*)+
for
1-x
sufficiently
_p(x,s) q
V)]
small
=
M
we have
for
Ap,
_%v (x'_*)M2s*x q
e
1 M2 M2s*x[ _ A i(s* 0 - --_ e i:O ,x)J
+
O(M4s*4(a
* (s,-l)
+ b
+
Bi(s
in M))
*
0 * ,x)Ji(s ,x)]
(B._3)
81
where
jO I
(s*_x)
=
]l_i
APov(_s) q
d_.
x This
to
Osborne's
representation
82
solution
thin
will
is
exact
to
airfoil
be
briefly
The
O(M2s*).
theory
[71].
discussed
The
in
first
term
usefulness
Appendix
C.
is
of
identical
this
APPENDIX
CALCULATION FOR
SIMPLE
In for
the
control
HARMONICALLY
this
appendix
calculation surfaces
will
of
solution
is
be
briefly
loads
two
methods.
well
the
known
for
loads
In
AND
methods on
LIFTING
and
wings
their
with
and
dimensional
extensions
without
Airfoils
incompressible
calculating
flow
pressure
Theodorsen's
distributions w
on
SURFACES
discussed.
Dimensional
-
LOADS
AIRFOILS
existing
aerodynamic
Two Analytical
OF AERODYNAMIC
OSCILLATING
some of
C
aerofoils.
For
a downwash
and
air-
n
V
= WnX
,
we have
Ap = -_-
Cp =
(
_ 4_
Wn[xn+(
n-i _ _=0
x n-_ -12-_ _ : I(_)
)+
ik(l+x) n+l
n
z xn-_ 2-_ _ : )+(C(k)-l) 2-I]i]: _:o,2,_,... T(_) _ ]
(C .i)
2 where
I(n)
=
n (5
n+l = -_
For
a down%ash
, ")
for
, n-i • V
given
by
n even , .
for
n odd.
w _-_(x - xc)n =
0
for
x
>
x C
for
x
M +
N_
equations
D.3
may
be
solved
for
the
matrices
Rj
and
Pk
successively. On
the
other
hand,
if F(z)
is known
for
z =
zk,
k =
0,1,2,3
... L
we have M F(Zk)Z
kN
=
N-I
_. i=O
PiZk i
F (Zk)
-
r. i:O
i,
R.z i
k = 0,I,2,3,...
L (D .4)
It
is now
known,
From
we
the
... M-l,
assumed may
PO
increase
L known and
that
value
values
rewrite
and
of
PM
are
known
apriori.
of M by
1 and
assume
Zk, we
equations
pick
D.4
M-1
in part
i=l
i:O
: i_
for
have
the
M-1
unknowns
unknowns P.
P.
and
1
in terms
M-I
of R.
1
is
PM+l
_,
not
=
[O]).
k I = 1,2,3,
as
N-I
now
that
values,
M-I
We
(If PM
equations
2,
and
3,
... M-I
hence
we
may
solve
as
1
N-1
Hence
equations
Pi
= J=O ?'
D.4
reduce
unknowns
Rj_
j = 0,1,2,
provided
the
system
also
possible
equations Pad_
are
to
solve
also
approximants
of
BijR'J +
to
are
discussed
of
which
equations
applicable
(D.6)
a system
... N-l,
equation
BO
are D.4
for
L-M+1
may
determine
solvable. directly
scalar
further
equations
On. the by
least
quantities. in
by
for
least other
the
N
squares hand
squares. Properties
it is
These of
[78].
99
Properties of Integral
Equations Related to Pad_Approximants ([79] and [80])
In this section someof the basic definitions
and properties
of
I
integral
equations
of
the
form
b _(x)
= f(x,s)
+
s J' dy K(x,y,s)_(y)
(D.7)
a
where are
f(x_s)
and
discussed.
K(x,y,s) We
can
are
defined
on
the
closed
internal
[a_b],
regard_
b K -- j' dy a in
equation
form
the
D-7
integral
as an
operator
equation
= f +
The
function
to ensure
a valid
restrictions and of
f are
can
K(x,y,s)
be
defined
acting
on
the
function
$.
In
operator
sK_
(D.9)
is the
imposed
and on
of bounded on
(D.8)
is
definition
functions
functions
K(x,y,s)(-)
the
_,
kernel
of
a unique f, and norm
inter_al
the
equation
solution K.
It
II "If in [a,bl;
for
of
and
D.7,
is usual the Hilbert
to
in order
various assume space
L2
example,
b
II 112=J'
.r-I
o
_-_ ._
.r-I 0
r.-.t
.el
CD
.,
144
LLb
t I I t
t tit lit Ill
I
Ill
l
L'-0 If
II
000 It
II
l|
4
,-..I
°r-f
.r.-i
._ c_
g °,-I
% 0 4_ o_
,--t .el °_
o;
0 0
-.,4
145
C) i
I
I
I
I
I
I
I
I
III
I
I
III
l
I I
III II
(i) 0
d
d
n
II
II
/
o_o II
II
II
/
/ -CD or) -0 i,m
0 f(._
o
! °_
0 -,-4 .._ .._ 0
E (1) of)
,-q CH
02
g (1)
.r-i
a_ > 0
0 4._
I..-
(1) ("xl C.) c" 0
4_ q_ .r-t ,--t ,-4
.r-I
"0 r-'t
.r"t
(Z3
I
I !
146
I
O3 0 r 0 I
E c°_
"U ,--t
c51 o
> 0
c5_ c5_ c5_ _%---.-
C) C ID
.H
r-4 o
H
_4 r-t
.r-t
i
I
i
I
!
147
0
,@ o
,_
*--4
.el
148
d
00
!-,,-cO
_ "ID I,,-,.
0 r!
E rd")
F:
0
C
Cl
r.t-i
I
0 ..p
I I
> U
I
I.
I I
0
-0,1 U'I
.r.I 0 -H
H
,.6 .el
_,lqp---__
__
149
f... (,/')
C_
..........Lcb 150
!
!
CD I
151
0
b..i
i
i'
.I
II
i-i 0 i--
I
+-. iW._ C_iO
i.--
"0 0
E-
I__ o o r_r_
v
0 r
,0. C_ C_. Ci II
_
0 II
IX') _
" .rq _YJ
.r-t % ©
,r'-t
,-t ,---t °r-t \
¢.) .,-t "u
,.d
\
\
2O
"ELastic Axis
2O 5O
Ref. Semi-chord: 11.985
Fig.
170
37.
Swept
wing,
AR
=
4.16,
A =
45 ° , M =
0.6
tO aJ X 0 ..Q
"o o
Or) X 0 r_
'0 ,.Q
aJ CO
-p .,-.I
-0 L_ 0 r o
C C] c__
o
o
Or) "0
to
f.--
4-_ o
0 ¢... 0
.o
I CO
,cl (L)
or)
,S o I--.
II
.el
r-
0 4_
°_
o
"0 a)
II
.x
H
,....-..e
(L) 0
% o
L.-
J
cO II
II
C) Z
C)
m
o °_
o tO .4--* (/1 "0
II
.r-.t ,r-I e)
u?
ci)
CD 0 o
CD
CD C_
.el
171
",4" @ O
O .H 0_ O
4_ -rH
O .H
F: r,.n c_ t
u3
C)
I,.,,.,-
! I
O
I
I ._
I I
-c,q E
nd
._
'-do O O
.m
@
LF_
ng C_J
O
C_ I,,,-.. O
O tCJ ug
uo CD C,O O
t72
_ ........ 2Lb
O
_
:
bid
_-t
-H
,r-i o
4_
H
_
-'..1"
0
E
I.0
0
0 cc.) Qm
I
-e,,Il:z l,/'l
0
> I,.. o "qt---"
o or.I
rD Q_
,'-t .el e,..l .el H
C_ 0
°_
c_ 0 Q
etb q 173
0
E Cl I..
O I
om
4_ Q.I 0
0 .rl
,m
.a-i i.--i
C_ CD CD
174
i.1" 0 C_
", U_
I I
O
I o4rJ
I I I
I
rom OJ
G.I > El i.
i,4 0 _-_
0 4_
G.I E El ._
I--t
,A
tO 41
, !
bO E----
CD Io
176
O
CD CD i•
C) C_ i °
1
o O_
E "tD I,--,,
0
o co
I I
I
I I
I
or)
i-t
"0 (ll
O_
(_
0
0 ._
I.-
(_ 0
i--I
C cJ
°_ ._ "0
Im H
"o
.r-t
I
CD
177
I
*-'I I
"0 L.
0 c" (_) I
© q-t
o °r"'l
..t.n
r...4 -_I °M
°_
l
0
c5
I
CO C_ 0
c5
178
"-.I"
|1 I
I I
I I
co
1 0 'k'_l
m
¢-. _-_
(1; 0
> 0
0 °_ 4J _J
U
C:
.r-I 0 .r-I _3
ID
:-6
I
I_0
1
,
-'0
o
0
c5 I
oaJD-_q-b_:uZ=_:_: c:; luawolg 179
rree
Vibration
Continuous Mode.
Anatysis
i
Loads
I......dl',,todeL For
1
I/
i i
IL-
' I
) l
FLutterAnatysis
I _J
Synthesis
{ Controt Law Synthesis
© Contro[
Laws
For
Flutter
Suppression
47.
Block diagram
for control
_U.S.
PRINTING
law synt_-esis procedure.
180 GOVERNMENT
OFFICE:
1977
- 73b-004/31
'
I
Dynamic Modet
Fig.
i
s.n.t,,_. I IA_o. L_,:,od_ 1
Shapes
't
For
