19840009069 NACA series
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Transcript of 19840009069 NACA series
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~ f f
lfI 35 8
S Technical Memorandum 8588
NASA TM 85881 1984 9 69
ifurcation nalysis of
ircraft
Pitching Motions Near the Stability
oundary
W H Hui and Murray Tabak
LI R RY py
January 984
NI\S \
National Aeronautics and
Space Administration
1
98
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RESE RCH CENTER
LIBR RY N S
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NASA Technical Memorandum 8588
ifurcation nalysis
of
ircraft
Pitching
Motions Near
the
Stability
oundary
W
H
Hui University
of
Waterloo Waterloo Ontario
Canada
Murray Tobak mes
Research
Center Moffett Field California
N \S \
National Aeronautics and
Space Administration
mes Research enter
Moffett Field California 94 5
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BIFURC TION
N LYSIS
OF
IRCR FT PITCHING
MOTIONS
NE R THE ST BILITY BOUND RY
W
H
Hui
Professor of
Applied
Mathematics
University
of Waterloo, Waterloo,
Ontario,
Canada
and
Murray Tobak
Research Scientis t
N S
mes
Research
Center, Mof fe tt
Field,
C
94 5
BSTR CT Bifurcation theory is used
to
analyze th e
nonlinear
dynamic
s t bi l i ty characteris t ics of
an
ircr f t subject
to s ingle-degree-of
freedom pitching-motion perturbations about a large mean
angle
of attack.
The requisite
aerodynamic
information in the equations of
mot ion can be
represented
in a form
equ ival en t to
th e response to
finite-amplitude
pitching
oscillations
about
the
mean
angle of
attack.
t
is
shown
how
this information can be
deduced
from the case
of
infinitesimal-amplitude
osci l lat ions.
The
bifurcation
theory
ana lysi s r evea ls
that when
the
mean angle of
attack i s increased
beyond a
cr i t ic l value at which the
aerodynamic damping vanishes, new
solutions
representing f in i te-
amplitude periodic motions
bifurcate from th e
previously
stable
steady
motion.
The
sign of
a
simple
cri terion,
cast in terms of aerodynamic
properties,
determines whether the bifurcating
s olu tions a re
stable
supercritical
or
unstable
subcrit ical .
For
f la t-plate
irfoils
flying
at
supersonic/hypersonic
speed,
the bifurcation i s subcri t ical ,
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implying
either th t
exchanges
of st bi l i ty
between
ste dy
and
periodic
motion re accompanied by hysteresis phenomena
or
t h t p ot en ti ll y l rge
periodic dep r tures
from
ste dy
motion
y
develop
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1.
INTRODU TION
Problems
of
aerodynamic stabi l i ty
of aircraf t
flying
at
small angles of
attack
have
been
studied ~ t n s i v l y
With
increasing
angles
of
attack
th e
problems
become more
complicated
and
typically
involve
nonlinear phenomena such as coupling between modes amplitude
and
frequency
effects , and hysteresis. The need
for inves tiga ting sta-
bi l i ty character is ti cs a t high
angles
of attack was
clearly
demonstrated
by Orlik-Ruckemann [1] in his survey paper which
largely
dea ls w ith
experiments.
On the
theoret ical
side, the greater part of an extensive body of
work i s
based
on the
l inearized
theory, in which
the
unsteady
flow is
regarded
as
a
small
perturbation
of
some known steady flow possibly
nonlinear
in ,
e g
the
angle of
attack
that
prevails under
certain
f l ight
conditions. The question
of
the validity and l imitations of such
a
l inearized
perturbation theory
is
of
fundamental importance
and
yet
has been investigated only
rarely.
One
may
argue
that
in
principle, i t
should be pos sib le t o advance to higher and higher angles of attack
by a
series
of l inear
perturbations, since the
solution at
each step
should include a steady-state part
which,
when added to the previous
s teady-s tate so lu t ion ,
would provide
the
s tart ing
point for
the next
perturbation. This
may
well
be
true provided that at
each
step
the
steady motion
is stable both stat ical ly
and dynamically,
and that
the
actual
disturbances,
e g
the
amplitude
of
oscil la t ion,
remain
small.
However the
l inear
perturbation procedure must
eventually cease
to be
valid
when th e angle of attack
exceeds
a certain cr i t i ca l value
cr
at
which the steady motion is no longer stable.
3
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In
this paper we investigate
the
st i l i ty characterist ics
of
an
4
ai rcraf t
trimmed to a mean angle
of
attack
near
m
cr
t
which the
steady
motion
becomes
unstable. Padfield
[2]
studied
a
similar
problem,
using the method of multiple
scales,
which is
valid only for weakly non-
l inear osci ll at ions . e shal l study th e problem by means
of
bifurcation
theory. This wil l allow
us
to draw on recent
mathematical
developments
e.g . ,
[3] that
are particularly
well s ui te d t o i nv es ti ga ti ng funda-
mental
questions in l inear and nonlinear
st i l i ty
theory. A
numerical
scheme based on bifurcation theory was proposed
ear l ier
[4] for
analyz-
ing
ircr f t
dynamic st i l i ty in a rather
general
framework More
recent
work [5 ]
demonstrates
the conside rable
potential
of bifurcation
theory in
f l ight
dynamics studies, particularly toward establishing a
method for th e design of f l ight control
systems
to ensure protection
against loss
of
control.
n
th e other
hand,
while acknowledging the
importance of the .aerodynamic model
in determining
the
i rc r f t s t i li ty
character is t ics , neither
of these
works
contains
an
adequate
assessment
of
the model requirements. The
t re atmen t o f
unsteady flow effects , in
particular, receives no at tent ion. In contrast, we shal l focus on just
this aspect of the problem
at
the expense of
narrowing
the
scope
of the
motion analysis. Restricting
the
motion to a
single-degree-of-freedom
pitching osci l la t ion
will enable us
to
analyze a motion for which com
plete aerodynamic information i s
available, for
certain aerodynamic
shapes, in the
form of
exact solutions of the inviscid supersonic/
hypersonic unsteady flow
theory
[6-10].
In this way will be possible
to establ ish a form revealing a precise analytical
relationship
between
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the basic
aerodynamic coefficients and
the
characteristics
of the
motion.
For a more extensive account of the analysis presented here, see [11].
2. M THEM TIC L
FORMUL TION
The Coupled Dynamic/Aerodynamic System.
res t r ic t
attention
to
the
sing1e-degree-of-freedom pitching
oscil la t ions of
an ai rcraf t
about a fixed
trim
angle. Before t ime zero,
le t the
ai rcraf t be in
level,
steady
f l ight
with
i t s
longi tudina l axi s inc lined
from
the
hori-
5
zontal
velocity vector by the fixed
trim
angle of attack o •
m
At
time zero
the
ai rcraf t
is
perturbed from i t s
trim
position,
but
during
the
subsequent
motion th e ce nte r of gravity
continues
to follow
a rect i -
l inear path at constant velocity
v
00
The
instantaneous angle of attack
o t
is
again
measured
relative to
the
horizontal velocity vector, while
the inclination of
o t from
the
fixed
trim angle
be
designated
s t .
The equations of
motion
are
0 - 0
dt m
I =
M t
dt
o
o t
- 0
wil l
m m
2.la
2.1b
where I
is
the moment
of inert ia
and
M t
th e
instantaneous aerody-
namic
pitching
moment both re fe rr ed t o th e
center
of gravity.
assume that
the
moment required to trim the
aircraft
at o has been
m
accounted
for,
so that M t is
a
measure of the perturbation
moment
only.
Now
we consider the
equations governing
the
flow
around the
aircraf t ,
assuming for simplicity
that
i t is permissible to neglect viscous effects .
The inviscid
unsteady flow-field
equations
are
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av Y
v • Vv
P
=
0
2.2a)
2.2b)
6
where
l L
~
•
v L
=
0 2. 2c )
at pY pY
p, p, and v
are
th e
pressure,
the density, and the velocity
vecto r, respec tively ,
and
y
is
the rat io of specif ic
heats
of the
medium Denote the posit ion vector originating from the center of
gravity by t and le t th e equation of the body surface be
B t ~ t » =
0,
which,
as no ted , depends
on th e
instantaneous displace-
ment angle The tangency condition at the body surface is
then
aB
v· • =
0
B
0
at v on =
2.3)
which also depends on The far-f ield boundary
conditions
relative
to a coordinate system fixed in the
aircraf t
are
2.4)
for subsonic f l ight or th e
well-known shock jump
conditions for
super-
sonic f l ight .
I t is clear from th e above
that
th e pitching motion of the
aircraf t
is coupled to the
unsteady flow
fie ld e .g ., the pressure
p)
through 2.3),
and, more direct ly ,
through
M t) in 2.la b),
which
is
determined from the instantaneous surface pressure
through
M t) =
VB
r x p
V
dS
S B o
2.5)
The
principal
di f f icu l ty however, is that the instan taneous surface
pressure in 2.5) depends not
only
on th e cur rent
state
of the
flow
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f ield
but also on i t s prior states . This means that past
values
of E
figure
in
the
determina tion of
the current state
of
the flow f ield and
M t
is
a
functional, not
a
fun ction , of
E;, t .
Thus,
unless approxi -
mations
are introduced,
a
determination of
E star t ing from given
in i t i l conditions
requires
the simultaneous solution of the coupled
equations
2.1-2.5 .
Although
the,
simultaneous solution of the coupled equations 2.1-2.5
i n p ri nc ip le
represents
an exact approach
to
the problem of determining
t ime-histories of
maneuvers
from given in i t i l conditions, i t i s
inevitably
a diff i ul t and
costly
approach see
t he d is cuss ion of [12] .
Approximate approaches
leading
to simpler,
le ss c os tly
computations are
a
pract ical necessity.
To
date, these
computations
have
invoked
the
assumption of a slowly varying
motion
of known form e.g. , a harmonic
motion whose aerodynamic force and moment
response
at t could
be
calculated.
This
stratagem, in .
ef fec t ,
uncouples the flow-field
equa-
t ions from the equations
of
motion. In a series
of papers see in
pa rt icu la r [13] , Tobak and
Schiff
have shown
how this
stratagem
can be
rationalized
to create mathematical models of the aerodynamic response
at
various levels of approximation of
th e
dependence on the past motion.
See also the contribution of Tobak , Chapman, and
Schiff
in
th is
collection.
Uncoupling
Near Neutral
Dynamic Stabil i ty Boundary.
Let
us
7
suppose that there
i s
a mean
angle of attack J
=
J
cr
t
which the
damping-in-pitch coefficient
vanishes. Then,
t and beyond this angle
of
attack,
the stead y
motion
of the ir r ft wil l no longer be
stable
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so that , in
response
to a disturbance,
the
motion
will seek a new
stable
s ta te which usually will consist of a f ini te-ampli tude periodic osci11a-
t ion
about
the
mean
angle of
attack.
The
changeover
from a
stable
steady motion to a
stable
periodic
motion
is called a Hopf
bifurcation
8
see
[3, 14], also
Sec. 4 . Thus, in the
vicini ty of
o where
the
cr
damping-in-pitch
coefficient
will
be
near zero, the consequent persis-
tence of oscil latory motions will make i t particularly appropriate to
assume a
periodic
past
motion
about a mean position in calculating the
aerodynamic
pitching
moment from the flow-field equations
for use
in
2.1 . This assumption
is
consistent
w ith the
mathematical
modeling
approach of Tobak and Schiff [13] at the second
level
of
approximation.
I t will be convenient to write M t in
the
form
2 - •
M t
=
p V
S ~ [ C ; ; 0
- C
0 ,0,0 ]
0000
m m m m
2.6
The
function
C ~ , t , o is the
pitching-moment coefficient
resulting
m m
from a
finite-amplitude periodic pitching motion about the center
of gravity,
and
C
0,0,0
is
i ts
steady-state
value at th e
mean
angle
m m
of attack
The function
C ~ , ~ , o depends on
the ins tantaneous
m m
displacement
angle i ts rate of change get , and the mean angle
of
attack
I t
also
depends, of cour se, on the
location
h of the
m
center of gravity re la t ive,
say,
to
the wing
leading edge,
the
f l ight
Mach
number
oo the ratio of
the
specific heats of
the medium
y,
and
the
aircraf t
shape.
A
great
amount
of
work
has been devoted
to the
theoretical determination of the function C for the
case
of
pitching
osci l lat ions of
infinitesimal
amplitude. For example , for the cases of
a wedge a f lat p late in supersonic/hypersonic
flow
[7,
8] and an ai r foi l
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of arbitrary profile· in
the
Newtonian l imit 19], this
function is
available
in exact analytical form for
large,
as well as small, mean
angles
of
attack
m and
includes th e
cr i t i ca l
angle
J
On the
cr
other hand, l i t t l e
is
known about the pitching-moment
coefficient
C
m
when the amplitude of oscil la t ion
is
f in i te
except for
the
special
case of a slowly oscil la t ing wedge
[10]. In the
next
section
we shall
show how th e func tion C for slow pitching
osci l lat ions of f in i te
m
amplitude may
be
obtained
from i t s behavior in the l imit of
osci l lat ions
of
infinitesimal amplitude.
The Pitching-Moment Function for Slow Pitching
Oscillations
9
of Finite
Amplitude.
.
Cons1der a
uniform
flow V past an aircraft
that
00
i s undergoing a slow pitching oscil la t ion
of f ini te
ampl itude about a
mean angle of
attack
m
The instantaneous
angular
displacement from
the
mean
position
is measured by set so that
l
si
is
the f ini te
max
amplitude of the oscil la t ion. I t i s required to calculate the unsteady
pressure f ie ld,
hence
the
form
of the
pitching moment M t in 2.1 .
Let the
cylindrical coordinates r , ~ , z
be such
that the z-axis
coin-
cides with the la te ra l
axis through
th e
center
of gravity.
Let
the
equation of the body
surface
at
the
mean angle of
attack
; A r,z . Then i t s
equation
at time t i s
m
B r , ~ , z , t
; A r,z set - ; 0
m
be
m
2.7
With velocity vector
becomes
v ; v , v ~ , v the boundary condi ti on 2 .3
r
I Z
2.8
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a t = A r,z cr
Equations 2.2 , 2,4 ,
and 2.8
complete
m
the
formulation for calculating th e flow
field in
terms
of
Now
for
a
long-established
periodic
motion
of
th e
a i r c r a f t ,
th e
aerodynamic
response
is also
periodic. Furthermore, invoking th e
assumption
of
slow
oscillations
allows us to
neglect
terms
of
0 g 2 , ~ and
higher
and
10
to w rite
P
=
p
1
V
=
V
0 l v
1
2.9
where
0
and ) 1 are
independent
of g t ) , but may depend on time
through
th e
function
Since
the zero th-order quantities 0
repre sent th e flow
field
when
=
0, such a flow must be quasi-steady;
i . e . , th e
time
variable
t can only appear implicitly through the
instantaneous
displacement
angle Hence,
Po
= po cr
m
l ; t , r , ~ , z , etc. 2.10
To
d erive the
mathematical
problem
for
th e f irs t-order quantities
) 1 we substi tute 2.9
into 2.2 , 2.4 ,
and 2.8 , use 2.10 , and
neglect
terms
of
· 2
••
O l
0
Thus,
2.l la
0 •
•
l
p
_ :2: VP =
1 1 0
Po
1
Po
0
o
2.11b
v
o
2.l lc
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and
a t
V
= r l V V
r z
=
A r,z)
+
a set)
m
2.11d)
2.11e)
11
Since the time variable t appears in 2.11) only as a parameter
through s e t ) ,
the
solution
to
2.11) must
be of
the form
2.12)
The f orms
of
p and p
o 1
in 2.10) and 2.12) determine
the
form
of
the
pressure p in 2.9)
which, in tu rn ,
determines
the
form of the
pitching-moment co efficien t a f t e r the integrations indicated in 2.5)
have been
carried
out. Thus
• s t
C s , s , o )
=
f o s) - - V g o s)
m m m m
00
2.l3a)
In the case of o s c i l l a t i o n s of in fin itesim al
amplitude,
2.13a
reduces
to
C
m
= f om) + sf om) + g om)
00
2.13b)
I t is
now clear
th at
the
functions
f o )
and g o ) are related to the
m m
s t i f f n e s s
der i vat i ve
S o ) and the
damping-in-pitch derivative
D o )
m m
a t
an angle
of
attack am as
defined
i n c la ss ic al aerodynamics, by
f o )
=
-S o ) ,
m m
g o ) = a )
m m
2.14)
Comparing 2.13a) and 2.13b), we
conclude
th at knowing the s t i f f n e s s and
damping
der i vat i ves
from the r e s u l t s
of
calculations for o s c i l l a t i o n s
of
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infinitesimal amplitude enables one to obtain immediately the pitching-
moment coefficient C for
the
corresponding finite-amplitude
case.
m
This
general
conclusion
i s
supported
by
the
form
of
th e exact
solution
for
C
for the
wedge
osc il la ti ng a t large amplitude
given
in
[10].
m
To
correct
a
misprint
in [10], a term
hA
4
[cos 6 - 6
a
- 1] should
be
added to t he r ight-hand side of Eq. 12b in [10].
Having determined an
appropriate
form
of M t for use in the
iner-
t i a l
equat ions o f
motion
2.1 ,
we
now
rewrite
th e equations introducing
2.l3a along
with
dimensionless time 1 e .
characterist ic
.lengths of
travel T = V t t Note that
we
shal l
retain the
symbol . to desig-
co
nate
a time derivative. Henceforth, however, the derivative will
be
with respect
to dimensionless
time:
.
= d/dT.
Let
F ~ ~ a = __
_ ~ T ~
__ = K[C - C
O O a
]
m
1 V /t 2
m m m
m
co
12
with
= K[f a
+
s - f a
+ +
s ]
m m m
2.15
P
St
3
co
K
21
•
ds
s -
dT
The
iner t ia l equations of
motion become
ds
•
S
dT
2.l6a
•
d F s,s,a
T
m
An
expansion of
F s ~ o
in
a
Taylor series in
m
2.l6b
sand s
and a
change of notation u
1
=
S
u
2
= s yields for 2.16
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13
i =
1,2)
2.17)
where
0
KD aJ
A=
-KS Om
B
1jk
a B
2jk
1
d
2
F
=
2
j ~
u=o
C
0
C
1 d
3
F
=
ljk£ 2jk£ 3
dUjdUkdU£
u=o
2.l8a)
2.l8b)
2.l8c)
Although
2.l6a b)
have
been
derived on the assumption of
slow oscil la-
tions terms in
C of
m
O g 2 , ~
omitted),
our subsequent bifurcation
analysis of 2.16)
wil l
hold fo r gene ral
F ~ , t , o
),
i . e .
as
i f
no
m
restr ict ion
had been placed on the
magnitude
of
t
In 2.18) the tensors
Band
C
represent
the effects of f ini te
amp1i-
tude
to
the second and third order. We note that the following symmetry
properties
hold:
2.19a)
2.19b)
On th e
basis
of
2.17), we
shall study the l inear
and
nonlinear stabi l i ty
of
the
motion in subsequent
sections.
3.
LINEAR
STABILITY THEORY. The stabi l i ty of th e steady motion at
an angle
of
attack
° to
infinitesimal disturbances
is determined by
m
the
nature of the
eigenvalues of the matrix A. They are
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3.1)
14
Case
I :
S om < O. In
t h i s
case .A
I
> 0 , A
2
< O. The steady motion
a t t hi s angle of
a tta c k
o
m
is
always unst abl e.
Case I I :
S o
> O.
m
Case I Ia : D Om <
O.
In th is case
Re A
I
;
0 an d
the steady motion
a t
0 is unst abl e.
m
Case
l Ib : D o )
> O.
m
a t
is
s t a b l e .
m
In
t h i s
case Re Al) < 0 an d the steady motion
2
Thus, only
in
Case l Ib, when both
s t i f f n e s s
and damping-in-pitch
d e ri va ti ve s a re p o s i t i v e , i s
the stead y
motion a t
angle of
attack
o
m
sta b le
to
i n f i n i t e s i m a l
di st ur bances. In f a c t , s tabi l i ty theory [3 ]
ca n
be
used to
show th a t stabi l i ty
o f the steady motion in
th is
case is
assured only i f t he d is tu rb a nc e is s u f f i c i e n t l y small.
In a l l
cases
in which
th e l inearized theory
predicts
growth of the
disturbance amplitude,
the
growth
predicted
is
o f e xp on en ti al
form
and
hence must
cease
to be valid af ter
some f in i te time when the
amplitude
i s no l on ge r s m al l. Thus, what eventually happens to a
motion
for which
l inearized s tabi l i ty theory predicts an in i t i a l growth
of
disturbances
cannot be determined from the l inearized theory i t se l f .
Instead,
the
full nonlinear
iner t ia l
equations
of
motion,
o r
a su ita b le approximation
of them,
such as
2.16) , must be
adopted
to
determine
the
ultimate s ta te
o f
the motion. Of
p a r t i c u l a r
in teres t is
th e
dynamic stabi l i ty
boundary
= o c r
where
S ocr)
>
0, D ocr) =
O.
The s tabi l i ty c h a r a c t e r i s t i c s
near t h i s
boundary w i l l be st udi ed in
th e
next section.
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4. NONLINEAR
STABILITY THEORY
At the dynamic stabil i ty boundary
a
=
a we
have S o
>
°
and D o =0
hence
c r c r cr
15
4.1)
The
existence
of
purely
imaginary e ig en va lu es o f
the matrix
A a t
a
=
a is the
c h a r a c t e r i s t i c
sign
of
a Hopf
bifurcation [3,
14],
m cr
signaling
a
changeover
from s table
steady motion t o p er io di c
motion.
On
crossing am
=
the steady motion
that
had
been
s table for
a
a w i l l become unstable t o d is tu rb an ce s, res ulting a f t e r a
m cr
trans ient
motion
ha s died away in
the
e xi st en ce o f a new motion, which
if
i t
is
stable) w i l l
be
periodic.
In the vicinity of
a
=
a the
m
c i r c u l a r frequency of
t he p er io di c motion
w i l l be
nearly
equal to wOo
cal l the
new solution
of the
e qu at io ns o f
motion a
b if ur ca ti on s ol u-
tion.
In t h i s
section we
s h a l l
determine i t s
character
and a c r i t e r i o n
fo r
i t s
s tabi l i ty .
Fo r
am
s l i g h t l y
larger
than
o c r
the
eigenvalues
of
the
matrix
A
are
Al
= - -
KD o ±
in o
m
where
s h a l l assume that
D 0
< °
r
4 . 2 )
4 . 3 )
4.4)
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which
i s
.the usual case in applications
[6].
The case
0
>
cr
can
16
be treated
in
exactly
th e same way.
The
normalized
eigenvector
1,; 0
m
associated
with
the
eigenvalue
A
0
is
m
4.5
whereas the adjoint eigenvector
s* o with eigenvalue
m
X o , which
is
m
the
complex conjugate
of
A
0
, i s
m
s* crm
4.6
A.
Hopf Bifurcation.
The bifurcat ion
solution
u . ,c r may
be
m
written as
-
- -
-
U =
a . s
+
4.7
Following
looss and
Joseph
[3] ,
p.
125 , we get
a
=
eb1 s
8
2
b
2
s
8
3
b
3
s
0 8
4
s = two
£2
w2
0 £4 ] .
am = Ocr
£2
0m2
0 £4
4.8
where, f or b re vit y, we omit the
lengthy
solution
forms
for £,
b
n
, w
2
,
and 0m2 cf.
[11] .
The
solution
is
periodic
in • with circular
frequency
equal to W
o
£2
w2
0 £4 .
B. Stabil i ty of the Periodic Bifurcation Solution. Accordingto
Floquet
theory
[3],
the stabi l i ty of t he p eri od ic bifurcat ion
solution
4.8 is
determined
by the sign of an index ll . To 0 £4 ,
has
the
form
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]l
4.9
17
and the
periodic bifurcation
solution
is stable i f ] l
0, unstable i f
] l > O
Since we have assumed
D cr )
0,
stabi l i ty
thus depends on
cr
the sign of 0m2 with
0m2 > 0 denoting stabi l i ty
and
0m2 < 0
insta-
bi l i ty .
I t
remains to
cast
0m2 in
more recognizable
terms. After
considerable
manipulation,
we get
4.10)
in terms of F U
1
,u
2
cr
m
.
From
2.15) we
see
that
the
function
F is
directly related to the pitching-moment coefficient
c
~ ~ c r )
acting
m m
on the aircraft
which is
performing a finite-amplitude
pitching
oscil la-
t ion around a mean
angle
of attack
a .
m
Equation
4.10)
demonstrates
that the stabi l i ty of the periodic motion
near
the dynamic stabi l i ty
boundary
a
cr
is
determined
by
the
behavior of the
aerodynamic response
c
~ ~ c r
) in
that
vicini ty.
m m
With th e
assumption of slow
oscillations
under
which the
form of
2.15)
was
derived
terms
of 0 t 2 , ~ neglected , we may substi tute
2.15)
into
4.10) to get
£2 2
02
f
= at at
4.11
From
2.14)
and the structure of the functions f and g, we write
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simple
form
4.13)
The
simplicity
of
t h i s re s u lt
suggests
t h a t
i t
might be
possible
to
derive
i t by
a less formal method.
This is indeed
the case:
have
v e rifie d
the
r e s u l t
by a physical approach f ami l i ar to workers in
the
f i e l d
o f n on li ne ar
m ech an ics.) Equation 4.13)
reveals
that
the sign of
and
thus
the
s tabi l i ty of the b i fu r ca t io n s o lu t io n , is independent of
th e
s c a la r
and
iner t ia l
pr oper t i es
of the ai rcraf t Rather, s tabi l i ty
depends only
on
whether the aerodynamic property DI /s is increasing
or decreasing
on crossing
the
dynamic
s tabi l i ty
boundary o
0
c r
The
two
p o s s i b i l i t i e s
are
w e ll i l lustrated
in th e
form
of bi furcat iondia
grams
as
shown in Figs. la and l b .
In
a
bifurcation diagram,
the
abscissa
r ep re se nt s t he parameter
that
is
being
var i ed, in
our case
the
mean
angle
of attack o •
The
ordinate
is a
parameter c h a ra c te ris tic of
the
bifurcation
s ol ut i on
alone. In
our
case
i t
is
a
measure
of
the
amplitude
of the
periodic
bifurcation
solution. Stable solutions
a re
indicated by solid
l i n e s , unstable
s ol u-
tions by
dashed
l i n e s .
Thus,
over the range of
mean
angle
of attack
o <
where the steady-state motion
is
s t a b l e ,
is
zero,
and
the
m
c r
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stable
steady motion is represented along th e abs ci ss a by a
solid l ine .
19
The steady
motion
becomes
uns table fo r a l l
values
of
o > 0 as the
m cr
dashed
l ine
along
t he abs ci ss a
indicates.
Periodic solutions
bifurcate
from 0 = 0 e i the r superc ri ti ca l ly or subcri t ical ly.
m cr
When (d/do
)(D }S)
_ > 0 (implying m2 >
0),
the bifurcation
m om-ocr
is called supercri t ical and
i t s
characteristic form is shown in Fig. lao
Stable
periodic
solutions
(solid
curves in Fig. la) exis t for values o f
o - 0 >
O
The amplitude
of the
periodic
solution
at a given value
m
cr
of
0 -
0 i s proportional to £ hence i s
vanishingly
small when
m cr
o -
0
is small,
varying
essentially
as 0 - 0
)1/2.
m cr m cr
When i o )(D /S) _ < 0 (implying m2 <
0),
the bifurcation
m om-ocr
is
called
subcri t ical and
i ts
characteristic form is shown in Fig. lb .
Periodic solutions
exis t for values o f 0 - 0 < 0, but they
are
m cr
unstable
(dashed curve in Fig.
lb) .
Whether stable periodic
solutions
do or do not exis t for o > 0 depends predominantly on the
behavior
m cr
of the
damping-in-pitch
derivative
D o for
m
o > 0 •
m
cr
If
no
such
stable periodic
solutions
exis t
for
o > then when
the
mean angle
m
cr
of
attack 0 i s increased beyond 0 the aircraft
may undergo an
m cr
aperiodic
motion
whose
departure
from the steady motion at
is potentially
large.
o = 0
m cr
In
the
more l ikely event th at s tab le
period ic solu tions
do exis t
for
o
>
0
(an
example
is
given
l a te r
their
amplitudes
must
be
f in i te
m
cr
and not infinitesimally
small, even
for small positive values of
m cr
I t
i s
l ikely that this branch of stable periodic so lu t ions
w ill join
that
of t he uns tabl e branch in
the
way i l luatrated in Fig.
lb.
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In
th is
event, the form of the
bifurcat ion curve for
values
of
<
m
cr
20
help s e xp la in
th e
s i tuat ion mentioned
ear l ier ,
where was
noted that
the steady-state motion could be stable to suff ic ient ly small
distur-
bances but become uns table t o larger disturbances. Thus, Fig. lb
suggests that for <
, s o long
a s d is tu rbance s are of small
m cr
enough
amplitude to
l i e below
those
of the
unstable branch o f
periodic
solutions curve
in
Fig. lb , they
wil l
die
out
with time and the
steady motion wil l remain s table . However d is tu rbance s w ith ampli -
tudes
suff ic ient ly
larger
than
those
of
t he uns tabl e
branch may
actually
grow up
to
th e u ltima te
motion
as
T , 00 ,
which
wil l
be
that
of
the
stable branch
of
pe ri od ic so lu ti ons curve
BAin
Fig. 1b . Finally,
we
note
that
i f
the
motion
does at tain the stable
branch of periodic solu-
tions say, for
<
then
hysteresis
effects
wil l
manifest them-
m
cr
selves with further changes in
m
When
m
is increased beyond
the motion wil l continue to be periodic with
f in i te amplitude
point A
in Fig . lb .
I f
is now
decreased
below , t h e periodic motion
m
cr
will p e rs is t,
even
a t values
of
where previously there had been
steady motion when
was being increased.
m
Not unt i l
i s dimin
ished beyond a
certain
point point B
in
Fig. lb
wil l
the motion return
to the
steady-state
c ondit ion poin t C in Fig. lb
that
had
been experi-
enced when
was increasing.
m
To further explore
the
implic ation s o f 4 .1 3 ,
we
invoke some approx-
imate relationships between the d m p i n g i n ~ p i t c h derivative
a
and
m
the
st i ffness
derivative S a . Tobak and Schiff have argued see,
in
m
part icular ,
[15] that , to good accuracy, a l inear
relationship
should
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exist between D and S at any value
of
a ;
i . e .
D o a - bS o
m m m
21
with b >
In
addition, we require that D vanish
a t
a
m
cr
The two
requirements
yield
the
form
D o
=
b[S o
-
S o ]
,
m
cr
m
b > 0
4.14
4.15
The validity of the approximate relation
4.14 has
been verified in
the
present study
via comparison with
exact resul ts in
[7]
for use
with osc il la t ing f la t-p la te
ai r foi l s
in supersonic/hypersonic flow
and
in [12] for use
with osci l lat ing
flaps in transonic flow. Replacing
D
in 4.13 by
4.14
casts
the
cr i ter ion solely in
terms
of S:
l = e:2. KS l/2.
b
[l.n
a
]
I
4
d0
2
m
m a =
m
cr
Thus,
near
a = a i
the
st if fness derivative S om increases
with
m
cr
a
slower/faster than exponen tial , the
periodic
bifurcation solution
m
is s tab le supercr it ica l /unstable subcri t ical . Cast in
terms of D
instead of S, the c ri te r ion s ta te s that
decreasing
D
with
respect to
slower/faster than exponenti al resul ts
in
stable supercr i t ical /
m
unstable subcritical periodic
bifurcation
solutions.
Examples
of
e
and D a
that
obey
4.14
and exhibit t he var ious
m m
possibil i t ies
are
a s f ollows .
Example
1:
= So exp[k o
- a
+
m o -
2]
}
m
cr
m
cr
=
bSo{l
-
exp[k a
-
m
-
2.]}
. m
cr
m
cr
4.16
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Example 2 :
22
= So exp[k a
- a
m a - a
)2 -
n a - a
)4] }
m
cr
m c r m
c r
=
bSo{l
-
exp[k a
-
a
+
m a -
a
)2
-
n
- a
t ]}
m
cr
m
c r
m
c r
4.17)
where
So
k ,
and n a re p os it iv e
constants. According
to
4.15),
m > 0
corresponds
to u nsta ble p eri od ic b i fu r c a ti o n s o lu t io n s
sub-
cr i t i ca l bi f ur cat i on) ,
and m <
0 corresponds to st abl e periodic
b i f u r -
c at io n s ol ut io n s
s u p e rc r it i ca l b i fu r c at io n ) .
In the case
of
s u b c r i t i c a l
bi f ur cat i on m> 0) in Example 1 , the
damping-in-pitch
derivative D1 a
m
continues to
decrease
to very
large
n eg ativ e v alu es as - 0
m c r
increases, which makes i t very unlikely t hat the
system
2.16) w i l l
have a
st abl e
periodic
motion
as a solution for
o > 0 •
m c r
On
the
o t h e r
hand,
in Example 2 th e
damping-in-pitch
derivative D
2
Om becomes posi -
t i ve
fo r s u f f i c i e n t l y large
-
I
m c r
so
t h a t
a st abl e periodic
motion may be
a
possible
solution
for
> 0 m > O.
m c r
Indeed, S2
an d
D
2
i n
Example
2 f ul f il l a ll of th e conditions s e t by
a
theorem of
Filippov [16],
the
s a t i s f a c t i o n
o f which
guarantees
t he e xi st en ce
of
a
st abl e periodic motion of
the
system 2.16) fo r
>
m
c r
As we
have
noted, r esul t i ng as i t
does from a
s u b c r i t i c a l bi f ur cat i on,
the solution
w i l l
e x hi bi t h y st er e si s
ef f ect s with
v a r i a t i o n s in
am below
c r
The
s t i f f n e s s
deriva-
5. SUPERSONIC/UYPERSONIC FLAT-PLATE
AIRFOILS.
To i l lust ra te
the
application
of b i f u r c a t i o n theory in
a
concrete
case,
we consider
a
f lat plate
a i r fo i l in supersonic/hypersonic
flow.
t iv e S a
m
an d the damping-in-pitch
derivative
D o a re known as
m
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analytic functions
of am
up to the ang1e-for-shock
detachment
[6-8].
They
depend on the f l ight
Mach number
M
w
the
rat io
of
specific
heats
y
h ere taken
to
be
that
of
a i r
y
=
1.4 , and
the
dimensionless
distance h of the center of gravity from the leading edge, here taken
as a fraction of the
chord
length
t
Results
are
presented in Fig. 2
of
log 8 0
versus a with
M
= 2.0, h = 0 and in Table 1
of the
m m
00
23
index
l 1
for various combinations of
M
and h.
00
I t
is shown
in Fig.
2
that
8 0
increases faster t han exponent ia l
near
a a 15.75°
m m cr
and
in Table
1 that l
is
always
posi t ive.
conclude that
whenever
the f la t -pla te a i r fo i l becomes dynamically unstable
[D o
= 0] the
cr
ensuing bifurcation always wil l be subcritica1.
Accordingly, th ere are two
possibi l i t ies.
One, a subcritica1 bifur-
cation
curve
such
as that
sketched in Fig.
1b exists , in which case the
a i r fo i l
motion wil l find
stabi l i ty at values
of
a > a
m
cr
in a
periodic
oscil lat ion
of f ini te amplitude.
At values
of
a a ,
the
m cr
steady-state
motion
wil l
be
stable
for
small
disturbances, but
for
larger disturbances
the
a i r fo i l
motion will
again seek the stable
periodic
motion. Exchanges
between
these two
stable
modes at
a < a
m cr
will be accompanied by hysteresis
effects .
Two al ternat ively, a bifur-
cation
curve
such as
that sketched
in Fig.
1b does
not
exis t , in which
case a
potential ly
large
aper iodic depar ture from the steady-state
motion
may
occur
as
a exceeds
m
a •
cr
In
ei ther case, on exceeding
the loss
of stabi l i ty
of the
steady-state
motion
must enta i l a
cr
discrete
change
to
a new
stable
condition.
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6. CONCLUDING REM RKS Bifurcation theory has been used to analyze
the nonlinear dynamic st bi l i ty characterist ics of an ircr f t
subject
to s i n g e ~ d e g r e e o f f r e e d o m pitching-motion
perturbations about a large
mean angle of attack. Setting up th e equations to which th e
bifurcation
theory was
applied
required, f i r s t determining conditions under which the
iner t i l equations of
motion and
th e gas-dynamic equations governing the
flow could be decoupled and,
second,
showing
how
the r equi red aerody
namic responses to f in i te -ampli tude
osci l lat ions
could be
obtained
from
the responses
to infinitesimal-amplitude
osci l lat ions.
Results of the
bifurcation
theory analysis revealed that
when
the
mean angle
of
attack
is
increased past th e cri t ic l
point
where the
aerodynamic
damping
vanishes,
new
solutions describing finite-amplitude
periodic motions bifurcate from t he previously
stable steady
motion.
The sign of a
simple
cri terion, cast in
terms
of aerodynamic properties,
determines whether the bifurcating solut ions a re stable supercrit ical
o r uns table
subcri t ical .
For
f la t-plate
irfoi ls
flying
t
supersonicl
hypersonic speed,
the
bifurcation
i s
subcri t ical ,
implying ei ther
that
exchanges of st bi l i ty
between
steady
and periodic
motions wil l
be
accompanied by
hysteresis phenomena
or
that a potentially dangerous
aperiodic
motion
may develop.
In
ei ther case the
loss
of st bi l i ty of
the steady-state motion must
be
accompanied
by a discrete change to a
new
stable
state.
4
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Acknowledgments
The
authors thank
Gary T. Chapman,
Ames Research
Center,
for
many
stimulating
discussions
during
the
course
of
this
research
and
Henry
J Van
Roese11
for helping w ith the
computations. W.
H. H. is
grateful
for the
financial
support
provided by
NASA Contract NAGW-130.
REFERENCES
1. K. J
Or1ik-Ruckemann, Dynamic
Stability Tes ting o f
Aircraft
Needs
Versus
Capabilities, Progress in Aerospace Sciences, Vol.
16,
No.4,
Pergamon
Press, New
York,
pp.
431-447
1975 .
2. G.
D.
Padfield, Nonlinear
Oscillations at High
Incidence,
G RD
CP-235, Dynamic Stabil i ty Parameters,
PaperNo.
31,
May
1978.
G
looss and
D. D. Joseph, Elementary Stability
and
Bifurcation
Theory,
Springer-Verlag,
New
York, 1980.
4. R. K. Mehra and J.
V.
Carroll,
Bifurcation
Analysi s of Aircraft
High
Angle-of-Attack Flight
Dynamics,
New
Approaches
to
Nonlinear Prob
lems
in
Dynamics, Ed.
P.
J.
Holmes,
SIAM
Philadelphia,
pp.
127-146
1980 .
5.
P. Guicheteau ,
Bifurcation
Theory
Applied
to the Study
of Control
Losses
on Combat
Aircraft , La
Recherche
~ r o s p a t i a l e pp. 1-14
1982-3 .
6.
w H. Hui, Stability
of
Oscillating
Wedges and
Caret
Wings
in
Hypersonic
and Supersonic Flows,
AIM J Vol. 7,
No.8,
pp.
1524-1530
Aug. 1969 .
7. W. H. Hui,
Supersonic/Hypersonic Flow
Past
an Oscillating
F1at
Plate
a t
High Angles
of
Attack, ZAMP
Vol. 29
Fasc. 3, pp.
414-427
1978 .
25
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8. W.
H.
Hui, An Analytic Theory
of
Supersonic/Hypersonic
Stabil i ty
at High Angles
of
Attack, G RD CP-235 Dynamic
Stabil i ty
Parameters,
Paper No. 22, May 1978.
9. W. H. Hui and
M.
Tobak Unsteady Newton-Busemann Flow Theory.
Part I : Airfoi ls , AIAA J Vol.
19,
No.3, pp. 311-318 Mar. 1981 .
10 . W
H.
Hui, Large-Amplitude Slow Oscillat ion
of
Wedges in
Inviscid Hypersonic and Supersonic Flows,
AlAA
J Vol. 8, No.8,
pp .
1530-1532 Aug. 1970 .
11 . W. H. Hui and
M.
Tobak
Bifurcation Analysis of Aircraft Pitching
Motions
About
Large
Mean
Angles of Attack.
To
appear
in
J.
Guidance,
Control, and Dynamics.
12 . W. J . Chyu and L. B. Schiff, Nonlinear Aerodynamic Modeling
of
Flap Oscillations in Transonic Flow: A Numerical Validation, AlAA J
Vol. 21 ,
No.1 ,
pp. 106-113
Jan.
1983 .
13 .
M.
Tobak and L. B.
Schiff ,
The Role of Time-History
Effects in
the Formulation of the Aerodynamics of Aircraft Dynamics G RD CP-235
Dynamic Stabil i ty
Parameters,
Paper No. 26 ,
May 1978.
14 . E. Hopf Abzweigung einer
periodischen
Losung von einer
Stationaren Losung eines Differentia1systems, Ber ichten der Mathematisch
Physischen K1asse
der Sachsischen
Akademie
der
Wissenschaften zu Leipzig,
Vol. XCIV, pp. 1-22 1942 .
15 . M. Tobak and L B Schiff, On
th e
Formulation of
th e
Aerodynamic
Characteristics in Aircraft Dynamics
NASA
TR R- 456, Jan. 1976.
16 . A.
F.Fil ippov, A
Sufficient Condition for the Existence o f
a
Stable
Limit
Cycle for an
Equation of the
Second
Order,
Mat.
Sbornik,
Vol. 30, pp . 171-180 1952 .
26
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•
Figure Captions
Fig.
1
Typical
forms
of bifurcation
diagrams
near the
dynamic
st i l i ty
27
boundary
where
P a
= O
•.
cr
c r
a)
Supercrit ical , d/da
)[D a
)/S a )] O
m m m am a
cr
b)
Subcrit ical ,
d/da
)[D a
)/S a )] < O
m m m am a
cr
Fig. 2
.stiffness
derivative
a versus mean
angle of attack
a for
m m
a
f l a t - p l a t e irfoi l
in
a
supersonic
free
stream:
y = 1 . 4 , K 1.
M 2 . 0 , h
0 ,
x
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8
Table
Values of stabi l i ty
cri ter ion ~ ~
h for f lat-plate irfoil
K £2 1 y 1.4;
>
subcritical bifurcation U <
supercritica1
bifurcation
h
h
0.1 0.2
0.3
0.4
0.1
0.2
0.3
0.4
00
00
1.5
36.5
19.0
2
39.5
25.8 17.2
13.0 14.7
1.6
39.6
23.4 13.6
8.8 10.7
3
57.2
38.6
26.8
20.9 23.3
1.7 40.0
24.8
15.5 11.0
12.8
4
84.7
58.4
41.4 32.8 36.4
1.8 39.7
25.3 16.3 12.0 13.7
5
107.3
74.8 53.7
42.9 47.5
1.9
39.4
25.5
16.8 12.6
14.3
6
123.6
86.7 62.6 50.3
55.6
2.0
39.5
25.8
17.2
13.0 14.7
h
0.25 0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.33
0.34 0.35
00
1.6 10.6
10.1 9.7 9.3 9.0 8.8 8.6
8.5 8.4
8.4
8.5
1.7
12.6
12.2 11.8
11.5 11.2
11.0
10.8
10.7 10.6 10.6 10.7
1.8
13.6
13.2
12.8 12.5
12.2 12.0 11.8
11. 7
11.6
11. 7 11. 7
1.9
14.1
13.7 13.4
13.0
.8
12.5
12.4
12.3
12.2 12.2
12.3
2.0
14.5
14.1 13.8 13.5 13.2 13.0 12.8
12.7 12.6 12.7 12.7
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1 2
E
o
;;; 1 0
w
. .
80
U-
60
40
~ _ - - -_-- . . L . . . _ L . . L . . . _ _
o 3 14 6 29 9 43 12 57 15 71 18 86 22 00
MEAN ANGLE OF ATTACK am
Fig.
2
30
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1
Report
No
NASA TM-8588l
2. Government Accession No. 3. Recipient s Catalog
No
4 Title and Subtitle
BIFURCATION
ANALYSIS
OF
AIRCRAFT
PITCHING MOTIONS NEAR THE
STABILITY BOUNDARY
5
Report Date
January 1984
6. Performing Organization Code
10. Work Unit
No
8. Performing Organization Report
No
A-9604
r : : : : : ~
7 Author(s)
W H
Hui
Univ.
o f
W a te r lo o , W a te r lo o ,
Ontario, Canada) and
Murray Tobak
9 Performing Organization Name and Address
T-4204
Ames Research
Center
Moffett F ield, CA
94035
,Contract or Grant No
1
13. Type of Report and Period Covered
12. Sponsoring Agency Name and, Address
National
Aeronautics and
Space Adm inistration
Washington,
20546
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
Originally
presented as a ~ t u r to th e Berkeley-Ames
Conference
on
Nonlinear
Problems in Contro
and
Fluid
Dynamics, Univ.
of
Calif or nia,
Berkeley,
Calif .
May
3l-June
10,
1983.
Point
o f C on ta c
Murray
Tobak, Ames Research Center,
MS
234-1, Moffett F ield, CA
94035 415)
965-5398 o r
FTS448-5398
16. Abstract
Bifurcation
theory is used to analyze t he n o nl in ea r dynamic stabi l i ty c ha ra c t e ri st i c s
of
an
aircraf t subject t o s i n gl e -d e g r ee - o f- f re e d o m p i tc h i ng - m ot i on perturbations
about
a large mean
angle
bf attack.
The re qui si t e
aerodynamic information·
in th e
equations
of
m ot io n c an be
repre
sented in
a form equivalent to
th e
response t o f in it e- am p li tu d e p it ch in g osc i l l a t i ons
about
th e
mean
angle
of
attack. I t
is shown how th is
information ca n be deduced
from th e case of
infinitesimal-amplitude
o s c i l l a t i o n s .
The
bif ur cation theory
analysis
reveals that
when th e mean
angle
attack is increased beyond a cr i t i ca l
value
a t which th e
aerodynamic
damping vanishes,
solutions
representing
f i n it e -a m p l it u d e p e ri o di c
motions bif ur cate from
th e
previously s table
steady motion. The sign
of
a simple c r i t e r i o n , cast in terms
of
aerodynamic p ro p er ti es , d e te r
mines whether th e bif ur cating so l u t i o n s
ar e
s table
s uper cr itical)
o r u ns ta bl e subc ri t i c a l ) .
Fo r
fL i t -pl a t e
airfoi ls
flying
a t
supersonic/hypersonic
speed,
th e
bif ur cation
is
s u b c r i t i c a l ,
implying e i t he r t h a t exchanges
of s tabi l i ty
between steady and
periodic
motion
are accompanied
by hys ter es is phenomena,
or
t h at p o te n ti a ll y
large
a p e ri o d ic d e p ar tu r es from steady motion may
develop.
17;
Key
Words (Suggested by Author(s))
Nonlinear aerodynamics
Nonst eady aerodynami cs
Large angles
of attack
18. Distribution Statement
Unlimited
Subject Category - 02
19. Security Classif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21. No. of Pages
33
22. Price*
D
*For sale by th e National Technical Information Service, Springfield, Virginia 22161
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