master's thesis defense improved system identification for ... · = solving for the x vector...
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Improved System Identification for Aeroservoelastic
Predictions
10:30 am2 July 2003
MAE Conference Room218 Engineering North
School of Mechanical and Aerospace EngineeringOklahoma State University
Abstract
Master's Thesis Defense
Charles Robert O'Neill
Modern high performance aerospace vehicles are particularly susceptible to destructive fluid-structure interactions. Accurate and timely aerodynamic predictions are needed for efficient vehicle design and evaluation. System identification offers an efficient and powerful prediction methodology by substituting a trained mathematical system model for the actual aerodynamic system. Coupling the system model with structural and control systems allows for fast and intuitive vehicle analysis. The challenge becomes determining a system model that accurately represents the dominant fluid-flow physics. This thesis investigated linear aerodynamic system identification for aeroservoelastic predictions based on Computational Fluid Dynamics (CFD) flow predictions.
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nb
ii
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ii ikxBikyAky
System Model
Excitation Signals
Aeroelastic Sensitivity Studies
1.141 1.0720.499
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Density Sweep Eigenvalues
Mach
Fresnel Chirp
Imp
rove
d S
yste
m Id
enti
ficat
ion
fo
r Aer
ose
rvo
elas
tic
Pred
icti
on
s
Sch
oo
l of M
ech
anic
al a
nd
Aer
osp
ace
Eng
inee
rin
gO
klah
om
a St
ate
Un
iver
sity
Mas
ter's
Th
esis
Def
ense
Ch
arle
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ber
t O
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llPr
esen
ted
by
2 Ju
ly 2
003
Aer
ose
rvo
elas
tici
ty Structure
Aerodynamics
Controls
Aeroelasticity
Aeroservoelasticity
ΣΣ
Input
Output
Aer
ose
rvo
elas
tici
ty is
a c
om
bin
atio
n o
f str
uct
ura
l, ae
rod
ynam
ics
and
co
ntr
ols
sys
tem
s.
Stru
ctu
ral S
yste
m
[]
[]
[]
Fq
Kq
Cq
M=
++
&&&
6 m
od
e p
late
exa
mp
le
[]
[]
[]
[]
)(
)(
)(
)(
)(
)1(
kF
Dk
xC
kq
kF
Hk
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kx
ss
s
ss
ss
+=
+=
+
The
dis
cret
e ti
me
stru
ctu
ral m
od
el is
th
e fo
llow
ing
:
The
gen
eric
st
ruct
ura
l g
ove
rnin
g
equ
atio
n
con
tain
s m
ass,
d
amp
ing
, st
iffn
ess
and
an
ex
tern
al
forc
ing
fu
nct
ion
.Gov
ern
ing
Eq
uat
ion
The
stru
ctu
ral
syst
em p
rop
erti
es a
re d
eter
min
ed
thro
ug
h d
eco
mp
osi
ng
th
e ov
eral
l st
ruct
ure
in
to
mo
des
hap
es a
nd
freq
uen
cies
.
Co
ntr
ols
Sys
tem
Det
erm
inin
g a
use
ful
con
tro
ls m
od
el
is c
on
cep
tual
ly s
trai
gh
tfo
rwar
d. T
he
exac
t im
ple
men
tati
on
met
ho
do
log
y ca
n v
ary.
A
gen
eral
ized
co
ntr
ol
law
fo
llow
s:
Structure
Aerodynamics
Controls
Aeroelasticity
Aeroservoelasticity
ΣΣ
Input
Output
[] x
Kr
=η
The
con
tro
l sys
tem
pro
per
ties
are
det
erm
ined
th
rou
gh
th
e d
esig
ner
's
spec
ific
sy
stem
p
erfo
rman
ce
crit
eria
. A
co
ntr
ol
syst
em
mo
difi
es t
he
over
all s
yste
m re
spo
nse
.
Ob
ject
ive
The
ob
ject
ive
is t
o d
evel
op
an
im
pro
ved
sys
tem
id
enti
ficat
ion
met
ho
do
log
y fo
r ae
rose
rvo
elas
tic
pre
dic
tio
ns.
Su
bst
itu
tin
g
the
tr
ain
ed
ae
rod
ynam
ic
syst
em
mo
del
fo
r th
e ac
tual
ae
rod
ynam
ic s
yste
m w
ill a
llow
fo
r an
eff
icie
nt
and
po
wer
ful p
red
icti
on
met
ho
do
log
y.
The
Pro
ble
mTr
adit
ion
al f
ree
resp
on
se a
nal
ysis
of
aero
spac
e ve
hic
les
wit
h
Co
mp
uta
tio
nal
Fl
uid
D
ynam
ics
req
uir
es s
ign
ifica
nt
com
pu
tin
g p
ow
er a
nd
do
es
no
t al
low
fo
r in
tuit
ive
or
qu
ick
anal
ysis
an
d
des
ign
.
Dec
om
po
se in
to 4
are
as
Aer
od
ynam
icSy
stem
Mo
del
Trai
nin
gM
eth
od
Exci
tati
on
Sig
nal
Perf
orm
ance
Cri
teri
a
Ho
w to
rep
rese
nt
the
aero
dyn
amic
sys
tem
Ho
w to
det
erm
ine
the
aero
dyn
amic
sys
tem
par
amet
ers
fro
m ra
w d
ata
Ho
w to
exc
ite
the
do
min
ate
aero
dyn
amic
s
Ho
w to
eva
luat
e an
d s
elec
t a
syst
em m
od
el
Un
stea
dy
Aer
od
ynam
ics
No
gen
eral
clo
sed
fo
rm s
olu
tio
n e
xist
s fo
r u
nst
ead
y ae
rod
ynam
ics.
Two
cl
assi
cal
un
stea
dy
aero
dyn
amic
exp
ress
ion
s re
sult
fro
m a
ssu
min
g a
n
inco
mp
ress
ible
, inv
isci
d f
low
wit
h s
imp
le m
oti
on
s an
d s
imp
le g
eom
etry
. Th
e W
agn
er, T
heo
do
rsen
an
d w
ave-
equ
atio
n e
xpre
ssio
ns
allo
w i
nsi
gh
ts
into
rep
rese
nti
ng
un
stea
dy
aero
dyn
amic
01
23
45
67
89
10
0
0.2
0.4
0.6
0.81
Ch
ord
s T
rave
led
Unsteady Lift Ratio
Ma
ch 0
.3
Ma
ch 0
.8
Ste
ad
y S
tate
Wa
gn
er
Wag
ner
Ste
p In
pu
t
Re
al C
(k)
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
0.20.10
0.1
0.1
0.2
0.3
0.5
1
24
∞0
Th
eo
do
rse
n F
un
ctio
n,
C(k
)
Re
du
ced
Fre
qu
en
cy
Uck
2ω=
Ima
gin
ary
C(k
)
Theo
do
rsen
Har
mo
nic
Mo
tio
nL
=πρb2
(h+
Uα−
baα)+
2πbρ
U(h
+U
α−
baα)C
(k)
Wav
e Eq
uat
ion
No
Mo
tio
nSu
bso
nic
Sup
erso
nic
D2
Dt2
=a2 0∇2
Aer
od
ynam
ic M
od
elin
g R
equ
irem
ents
Co
nsi
sten
t B
ou
nd
ary
Co
nd
itio
ns
Inp
ut-
Ou
tpu
t D
ynam
ics
Star
tin
g a
nd
En
din
g C
on
dit
ion
s
Acc
ura
cy n
ear s
yste
m s
tab
ility
po
int
Cap
ture
s d
iffer
ent
tim
e sc
ales
Mo
tio
n li
mit
atio
ns
to re
mai
n in
lin
ear r
egio
n
[](
)[]
∑∑
− ==
−+
−=
1 01
)(
)(
nb ii
na ii
ik
qB
ik
fA
kf
Inte
rnal
Resp
onse
Inpu
t Res
pons
e
Aer
od
ynam
ic R
epre
sen
tati
on
s
f(n
)+
...+
f+
f+
f+
f(d
elays)
=x
(n)+
...+
x+
x+
x+
x(d
elays)
+C
Usi
ng
th
e p
revi
ou
s u
nst
ead
y ae
rod
ynam
ic
resu
lts
and
m
od
elin
g
req
uir
emen
ts, t
he
follo
win
g g
ener
ic a
ero
dyn
amic
fun
ctio
n is
pro
po
sed
.
No
w, a
sp
ecifi
c sy
stem
mo
del
nee
ds
to b
e se
lect
ed. T
he
bes
t ca
nd
idat
es
wer
e th
e in
dic
ial r
esp
on
se m
od
el a
nd
th
e A
RMA
mo
del
.
Ind
icia
l Res
po
nse
ARM
A M
od
elTh
e A
uto
Reg
ress
ive
Mov
ing
Ave
rag
e (A
RMA
) mo
del
co
nsi
sts
of a
n in
pu
t-o
utp
ut
exp
ress
ion
w
ith
in
tern
al
and
in
pu
t re
spo
nse
s.
Ind
ivid
ual
co
effic
ien
ts A
i an
d B
i det
erm
ine
the
syst
em p
aram
eter
s. Th
e A
RMA
mo
del
d
irec
tly
corr
esp
on
ds
to a
dis
cret
e-ti
me
OD
E re
pre
sen
tati
on
.
The
ind
icia
l res
po
nse
co
nsi
sts
of
a st
ep r
esp
on
se. T
he
syst
em r
esp
on
se is
d
eter
min
ed t
hro
ug
h c
onv
olu
tio
n. T
he
syst
em p
aram
eter
s ar
e d
eter
min
ed
by
the
step
re
spo
nse
fu
nct
ion
g
(t).
Det
erm
inin
g
g(t
) b
eco
mes
co
mp
licat
ed w
hen
th
e b
ou
nd
ary
con
dit
ion
s ar
e co
up
led
.
y(t
)=
∫ t 0
g(t−
τ)u
(t)d
τ
Trai
nin
g M
eth
od
olo
gy
The
trai
nin
g m
eth
od
red
uce
s th
e ra
w a
ero
dyn
amic
tim
e h
isto
ries
to
usa
ble
aer
od
ynam
ic s
yste
m m
od
els.
Re
ad
xn.d
at
De
tre
nd
SV
D o
ne
ach
mo
de
Pa
cka
ge
into
.m
dl
FA
XN
FA
OF
FS
ET
S
A, B
This
is a
lin
ear e
qu
atio
n o
f th
e fo
rm:
Exp
ress
ing
th
e in
pu
ts a
nd
ou
tpu
ts in
th
e A
RMA
sys
tem
form
yie
lds:
[]
bx
Ar
r=
Solv
ing
fo
r th
e x
vect
or
yiel
ds
the
AR
MA
mo
del
co
effi
cen
ts.
Sin
gu
lar
valu
e d
eco
mp
osi
tio
n (
SVD
) is
use
d t
o s
olv
e th
is l
inea
r le
ast
squ
ares
alg
ebra
ic s
yste
m. S
VD
is
pre
ferr
ed b
ecau
se o
f it
s ro
bu
stn
ess
and
its
abili
ity
to s
olv
e ov
erd
eter
min
ed s
yste
ms.}
Out
put
Cur
rent
In
puts
Cur
rent
and
Past
nr1
Out
puts
Pr
evio
us
)(
)1(
)1(
)(
)(
)(
)1(
M
44
44
44
44
44
48
444
44
44
44
44
76
ML
M
LL
L4
44
48
444
47
6
MLt
y
BAn
bt
-x
nb
t -
xt
xt
xn
at
-y
t -
y
iin
r1
- -
[[=
[[
[
[
Para
llel T
rain
ing
Para
llel
trai
nin
g i
s p
oss
ible
by
mak
ing
no
tici
ng
th
e st
ruct
ure
of
the
trai
nin
g d
ata
mat
rix.
Th
e co
lum
ns
rep
rese
nt
the
mo
del
ord
er, t
he
row
s re
pre
sen
t th
e in
div
idu
al t
imes
tep
s.
Cat
enat
ing
ad
dit
ion
al r
ow
s o
nto
th
e m
atri
x is
per
mit
ted
. Th
e o
nly
cav
eat
is t
o e
nsu
re c
on
sist
ency
of
pre
vio
us
forc
es a
nd
dis
pla
cem
ents
occ
uri
ng
b
efo
re t
ime
zero
.
Ad
van
tag
es:
Elim
inat
es lo
ng
-lag
co
nta
min
atio
nEl
imin
ates
mo
dal
dis
per
sio
nD
istr
ibu
ted
co
mp
uti
ng
Exci
tati
on
tai
lori
ng
Le
ss s
ensi
tive
to e
rro
rs t
han
s
eria
l tra
inin
g
Dis
adva
nta
ges
:Ex
tra
bo
okw
ork
Mu
st jo
in a
ll si
gn
als
for m
od
el e
valu
atio
ns
y(0
)x(1
)x(2
)y(1
)x(2
)x(3
)y(2
)x(3
)x(4
)
·
A1
B1
B2
=
y(1
)y(2
)y(3
)
Exci
tati
on
Sig
nal
sFo
r th
e ae
rod
ynam
ic id
enti
ficat
ion
pro
cess
to
cap
ture
a
use
ful
mo
del
, th
e d
om
inan
t ae
rod
ynam
ics
mu
st b
e ex
cite
d. T
he
exci
tati
on
is
dir
ectl
y re
late
d t
o t
he
inp
ut
sig
nal
. Th
e fo
llow
ing
cr
iter
ia
wer
e id
enti
fied
as
im
po
rtan
t to
su
cces
sfu
l in
pu
t si
gn
al d
esig
n. T
he
crit
eria
ar
e b
ased
on
ph
ysic
s, p
erfo
rman
ce a
nd
sys
tem
mo
del
ch
arac
teri
stic
s.
Cri
teri
aTh
e ex
cita
tio
n m
ust
be
con
sist
ent
wit
h t
he
un
stea
dy
aer
od
ynam
ic re
pre
sen
tati
on
.
Stat
ic o
ffse
t fo
rces
mu
st b
e ca
lcu
late
d
Exci
te t
he
do
min
ant
un
stea
dy
aero
dyn
amic
s w
hile
bei
ng
kep
t in
th
e "l
inea
r" a
ero
dyn
amic
ran
ge
Inp
ut
and
ou
tpu
t d
ynam
ics
mu
st b
e ex
cite
d
Exci
te t
he
syst
em w
ith
in a
use
ful f
req
uen
cy ra
ng
e
Mu
ltis
tep
01
02
03
04
05
06
07
08
09
01
00
10
4
10
2
10
0
10
2
10
4
Dis
pla
cem
en
t
PSD
01
02
03
04
05
06
07
08
09
01
00
10
2
10
0
10
2
10
4
Fre
qu
en
cy (
%N
yqu
ist)
Ve
loci
ty
PSD
20
24
68
10
10.50
0.51
Ve
loci
ty
20
24
68
10
1
0
0.51
Dis
pla
cem
en
t
0.5
Ad
van
tag
es:
Easy
to im
ple
men
tSi
mp
le fu
nct
ion
al fo
rmB
road
PSD
Co
mm
on
flig
ht-
test
sig
nal
Dis
adva
nta
ges
:In
con
sist
ent
Bo
un
dar
y C
on
dit
ion
sN
on
intu
itiv
e ex
cita
tio
n le
ng
thH
ole
s in
PSD
Ch
irp
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
1
0.
50
0.51
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
21
012
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
42
024
Dis
pla
cem
en
t
Ve
loci
ty/w
Acc
ele
ratio
n/w
2
01
02
03
04
05
06
07
08
09
01
00
10
4
10
2
10
0
10
2
Dis
pla
cem
en
t
PSD
01
02
03
04
05
06
07
08
09
01
00
10
2
10
0
10
2
10
4
Fre
qu
en
cy (
%N
yqu
ist)
Ve
loci
ty
PSD
Lin
ear F
req
uen
cy S
wee
p
d(t
)=
sin(
ωt2
)
v(t
)=
2ωtco
s(ωt2
)
Ad
van
tag
es:
Co
nsi
sten
t w
ith
fun
ctio
nal
req
uir
emen
tsFl
at P
SDIn
tuit
ive
exci
tati
on
len
gth
Dis
adva
nta
ges
:Po
or l
ow
freq
uen
cy p
erfo
rman
ceO
nly
cap
ture
s an
alyt
ic re
spo
nse
Can
ove
rexc
ite
the
syst
em
DC
Ch
irp
Lin
ear F
req
uen
cy S
wee
p w
ith
a D
C O
ffse
t0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10
0.51
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
1
0.
50
0.51
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
21
012
Dis
pla
cem
en
t
Ve
loci
ty/w
Acc
ele
ratio
n/w
2
01
02
03
04
05
06
07
08
09
01
00
10
10
10
5
10
0
10
5D
isp
lace
me
nt
01
02
03
04
05
06
07
08
09
01
00
10
2
10
0
10
2
10
4
10
6
Fre
qu
en
cy (
%N
yqu
ist)
Ve
loci
ty
d(t
)=
−1 2
cos(
ωt2
)+
1 2
v(t
)=
ωtsi
n(ωt2
)
Ad
van
tag
es:
Sam
e ad
van
tag
es a
s th
e ch
irp
Imp
rove
d lo
w fr
equ
ency
exc
itat
ion
Dis
adva
nta
ges
:Pe
ak fa
cto
r is
twic
e th
e ch
irp
's
Fres
nel
Ch
irp
The
Fres
nel
ch
irp
is t
he
inte
gra
lo
f th
e si
ne
fun
ctio
n. T
he
Fres
nel
form
is c
om
mo
nly
see
n in
op
tics
.
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
1
0.
50
0.51
00
.10
.20
.30
.40
.50
.60
.70
.80
.91
0
0.2
0.4
0.6
0.8
Dis
pla
cem
en
t
Ve
loci
ty
01
02
03
04
05
06
07
08
09
01
00
10
4
10
2
10
0
10
2
10
4
Dis
pla
cem
en
t
PSD
01
02
03
04
05
06
07
08
09
01
00
10
4
10
2
10
0
10
2
Fre
qu
en
cy (
%N
yqu
ist)
Ve
loci
ty
PSD
v(t
)=
sin(
ωt2
)
d(t
)=
S(t
)=
∫si
n(ωt2
)
Ad
van
tag
es:
Flat
PSD
for v
elo
city
DC
off
set
com
po
nen
t
Dis
adva
nta
ges
:N
o c
lose
d fo
rm s
olu
tio
nD
isp
lace
men
t PS
D h
as h
ole
sO
ffse
t Fr
esn
el is
imp
ract
ical
Sch
roed
er S
wee
p
05
10
15
1
0.
50
0.51
Dis
pla
cem
en
t
05
10
15
30
20
10
0
10
20
30
Ve
loci
ty
01
02
03
04
05
06
07
08
09
01
00
10
6
10
4
10
2
10
0
10
2D
isp
lace
me
nt
PSD
01
02
03
04
05
06
07
08
09
01
00
10
2
10
0
10
2
10
4
Fre
qu
en
cy (
%N
yqu
ist)
Ve
loci
ty
PSD
Ad
van
tag
es:
Flat
PSD
Op
tim
al p
eak
fact
or
Des
irab
le lo
w fr
equ
ency
resp
on
seH
arm
on
ic s
ign
al a
llow
s an
arb
itra
ry
e
xcit
atio
n le
ng
th
Dis
adva
nta
ges
:D
oes
no
t st
art
fro
m re
stSe
nsi
tivi
ties
to e
xcit
atio
n m
eth
od
The
Sch
roed
er fo
rm is
bas
ed o
na
sum
of c
osi
ne
term
s w
ith
asp
ecifi
ed p
has
ing.
Th
e fo
rm is
anal
ytic
in t
ime
bu
t n
ot
smo
oth
in fr
equ
ency
.
d(t
)=
N ∑ k=
1
√1 2N
cos(
2πkt
T−
πk
2
N)
v(t
)=
N ∑ k=
1
−2πk
T
√1 2N
sin(
2πkt
T−
πk
2
N)
No
ise
An
oft
en p
rop
ose
d s
olu
tio
n is
to u
sea
no
ise
inp
ut
sig
nal
. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11
0.
50
0.51
01
02
03
04
05
06
07
08
09
01
00
10
3
10
2
10
1
10
0
10
1
PSD
10
6
10
4
10
2
10
0
10
2
PSD
Ad
van
tag
es:
Flat
PSD
No
lim
it o
n e
xcit
atio
n le
ng
thEx
cite
s en
tire
flo
w d
ynam
ics
Dis
adva
nta
ges
:Fl
at P
SD o
nly
occ
urs
in t
he
limit
No
n in
tuit
ive
exci
tati
on
len
gth
No
n d
eter
min
isti
c ca
use
s B
C p
rob
lem
sEq
ual
po
wer
mea
ns
"sh
arp
" ch
ang
es
PSD
for 5
00 a
nd
500
0 d
ata
po
ints
Mo
tio
n
Art
ifici
al N
ois
e0
0.5
11
.52
2.5
33
.5
05
10
No
isy
1
00
.51
1.5
22
.53
3.5
05
10
1
Cle
an
01
02
03
04
05
06
07
08
09
01
00
10
30
10
2
5
10
2
0
10
1
5
10
1
0
10
5
10
0
10
5
10
10
10
15
Tran
sfer
Fu
nct
ion
Ou
tpu
t
Inp
ut
Freq
uen
cy (%
Nyq
uis
t)
Pow
er S
pec
tral
Den
sity
01
02
03
04
05
06
07
08
09
01
00
10
14
10
1
2
10
1
0
10
8
10
6
10
4
10
2
10
0
Pow
er S
pec
tral
Den
sity
Tran
sfer
Fu
nct
ion
Ou
tpu
t
Inp
ut
Freq
uen
cy (%
Nyq
uis
t)
No
isy
Sig
nal
Cle
anSi
gn
al
Co
ntr
ary
to in
tuit
ion
, ad
din
g n
ois
e to
an
exis
tin
g s
ign
al is
oft
en d
esir
ed fo
r bet
ter
hig
h fr
equ
ency
mo
del
ing.
Mo
tio
n S
pec
ifica
tio
n
Stri
ct S
pec
ifica
tio
n
Stat
e Sp
ace
Spec
ifica
tio
nA
dva
nta
ges
:Si
mu
ltan
eou
s B
C u
pd
ates
Inp
ut
sig
nal
s ar
e fil
tere
d t
hro
ug
h a
"
stru
ctu
re"
wit
h m
ass.
Dis
adva
nta
ges
:Re
qu
ires
sel
ecti
ng
a "
stru
ctu
ral m
ass"
Un
der
sira
ble
sto
pp
ing
co
nd
itio
ns
lead
to m
oti
on
s th
at a
re n
on
linea
r.
Ad
van
tag
es:
"Per
fect
" si
gn
als
Easy
imp
len
tati
on
Dis
adva
nta
ges
:N
ot
con
sist
ent
in a
dis
cret
e ti
me
sen
se
For
stri
ct
mo
tio
n
spec
ific
atio
n,
the
mo
tio
n
bo
un
dar
y co
nd
itio
ns
at
each
ti
mes
tep
ar
e d
eter
min
ed
fro
m
a n
um
eric
al o
r an
alyt
ical
exp
ress
ion
.
For
stat
e sp
ace
mo
tio
n s
pec
ifica
tio
n, t
he
bo
un
dar
y co
nd
itio
ns
are
up
dat
ed
sim
ult
aneo
usl
y th
ou
gh
th
e m
oti
on
sta
te
vect
or.
This
exa
ctly
du
plic
ates
th
e ac
tual
d
iscr
ete
tim
e C
FD m
oti
on
form
.
Mo
del
Eva
luat
ion
An
eva
luat
ion
met
ho
do
log
y is
nee
ded
to
co
mp
are
and
sel
ect
the
"bes
t"
aero
dyn
amic
sys
tem
mo
del
. Vis
ual
insp
ecti
on
is d
oo
med
; th
e ev
alu
atio
n
mu
st b
e b
ased
on
a n
um
eric
al m
easu
rem
ent.
Eval
uat
ion
bas
ed o
n M
od
el E
rro
rs
χ2
=||A
·x−
b||2
nst
p
02
04
06
08
01
00
12
01
40
16
01
80
20
01
06
10
4
10
2
10
0
10
2
Nu
mb
er
of
Mo
de
l Co
eff
icie
nts
c2
02
04
06
08
01
00
12
01
03
10
2
10
1
Nu
mb
er
of
Mo
de
l Co
eff
icie
nts
Fo
rce
RM
S
Ind
icat
es t
he
mat
rix
solu
tio
n q
ual
ity
Ind
icat
es t
he
mo
del
pre
dic
tio
n q
ual
ity
RM
S=
√ √ √ √1 N
N ∑ k=
0
(y(k
)−
y(k
))2
Mo
del
Eva
luat
ion
Eval
uat
ion
bas
ed o
n A
ero
elas
tic
Stab
ility
Usi
ng
th
e m
od
el f
or
cou
ple
d a
ero
stru
ctu
ral
pre
dic
tio
ns
exer
cise
s th
e m
od
el's
pre
dic
tio
n c
har
acte
rist
ics
in a
rea
listi
c an
d i
ntu
itiv
e m
ann
er.
This
eva
luat
atio
n i
s b
ased
on
sys
tem
eig
enva
lue
pre
dic
tio
ns.
Th
e co
up
led
aer
oel
asti
c p
lan
t m
atri
x is
:
05
10
15
20
25
30
0.4
0.4
5
0.5
0.5
5
0.6
0.6
5
0.7
0.7
5
0.8
12
3456
7891
01112131415161718192
nb
Ord
er
Dynamic Pressure
na
Ord
er
Eig
enva
lue
Form
Stab
ility
Bo
un
dar
y Fo
rm
[ Gs+
q ∞H
sD
aC
sq ∞
HsC
a
HaC
sG
a
]
AG
ARD
445
.6
Two
Mod
e St
ruct
ural
Mod
el
Firs
t Ben
ding
Mod
e9.
6 H
ertz
Firs
t Tor
siona
l Mod
e38
.2 H
ertz
45 d
egre
e sw
eep
Asp
ect R
atio
260
% ta
per r
atio
NAC
A 6
5A00
4 Ro
ot
Tip
V
AG
ARD
445
.6
05
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
1
23 4 56
7 8 91
0 1112131415 1617181920
nb
Ord
er
Dynamic Pressure
na
Ord
er
05
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.91
0
12
3456
7891
011121314151617181920
nb
Ord
er
Dynamic Pressure
na
Ord
er
05
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
12
3456
7891
011121314151617181920
nb
Ord
er
Dynamic Pressure
na
Ord
er
05
10
15
20
25
30
0
0.2
0.4
0.6
0.81
1.2
1.4
1.6
1.8
01
2
3 456 7 8 910111213 14 15161718 1920
nb
Ord
er
Dynamic Pressure
na
Ord
er
01
02
03
04
05
06
00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1 234567891011121314151617181920
nb
Ord
er
Dynamic Pressure
na
Ord
er
05
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
2
3456
78 910
nb
Ord
er
Dynamic Pressure
na
Ord
er
Ch
irp
DC
Ch
irp
DC
Ch
irp
Larg
e A
mp
litu
de
Mu
ltis
tep
Fres
nel
wit
h
Stat
e Sp
ace
Spec
ifica
tio
nSc
hro
eder
AG
ARD
445
.6
0.4
0.5
0.6
0.7
0.8
0.9
11
.11
.20
0.2
0.4
0.6
0.81
1.2
1.4
1.6
1.8
Ma
ch N
um
be
r
Dynamic Pressure [psi ]
Stab
ility
Bo
un
dar
y
AG
ARD
445
.6 S
ensi
tivi
ty S
tud
ies
Eig
enva
lues
in z
-plan
e at
Mac
h 0.
499
Uni
t Circ
le, |
z|=
1
Mod
e 2
Mod
e 1
Incr
easin
gD
ensit
y
Incr
easin
gD
ensit
y
|z|=
1
1.14
11.
072
0.49
9
0.67
80.
900
0.96
0
Den
sity
Swee
pM
odel
Sens
itivi
ty S
tudy
0.4
0.5
0.6
0.7
0.8
0.9
11
.11
.20
0.2
0.4
0.6
0.81
1.2
1.4
1.6
1.82
Ma
ch N
um
be
r
Sta
bili
ty B
ou
nd
ary
[p
si]
+10
%
-10%
As-
Is
Str
uct
ura
l Fre
qu
en
cy
Stuc
tura
l Fre
quen
cy S
tudy
02
46
81
01
21
41
60
.7
0.7
5
0.8
0.8
5
0.9
0.9
51
1.0
5
1.1
Da
mp
ing
Ra
tio %
Sta
bili
ty B
ou
nd
ary
[p
si]
Ma
ch 0
.49
9
Mach
1.0
72
Stuc
tura
l Dam
ping
Stu
dy
Win
g/F
lap
Co
ntr
ol
U
θδ
De
L
Sch
emat
ic
CFD
Gri
d
05
10
15
20
25
30
35
0
0.51
1.52
Dis
pla
cem
en
t
1 2
05
10
15
20
25
30
35
20
10
0
10
20
Ve
loci
ty
1 2
05
10
15
20
25
30
35
0.0
1
0.0
050
0.0
05
0.0
1F
orc
es
12
05
10
15
20
25
30
35
0
0.51
1.52
Dis
pla
cem
en
t
12
05
10
15
20
25
30
35
20
10
0
10
20
Ve
loci
ty
12
05
10
15
20
25
30
35
2024x
10
4F
orc
es
1 2
Trai
nin
g S
ign
als
Win
g/F
lap
Co
ntr
ol
Op
en L
oo
p
00
.01
0.0
20
.03
0.0
40
.05
0.0
60
.07
0.0
80
.09
0.1
40
20
0
20
40
Dis
pla
cem
en
t
12
00
.01
0.0
20
.03
0.0
40
.05
0.0
60
.07
0.0
80
.09
0.1
2101x
10
4V
elo
city
12
00
.01
0.0
20
.03
0.0
40
.05
0.0
60
.07
0.0
80
.09
0.1
500
50
Fo
rce
s
1 2
00
.10
.20
.30
.40
.50
.60
.70
.85
05D
isp
lace
me
nt
12
00
.10
.20
.30
.40
.50
.60
.70
.81
00
0
50
00
50
0
10
00
Ve
loci
ty
1 2
00
.10
.20
.30
.40
.50
.60
.70
.82024
Fo
rce
s
12
Clo
sed
Lo
op
k={3
.5, 0
, -0.
01, -
0.04
}
Clo
sed
Lo
op
k={3
.5, 0
, -0.
01, -
0.05
}
Clo
sed
Lo
op
Eig
enva
lues
k={3
.5, 0
, -0.
01, -
0.05
}
Eig
enva
lues
nea
r th
eu
nit
cir
cle
are
cau
sin
ga
no
n-p
hys
ical
inst
abili
ty.
Co
ncl
usi
on
sIm
pro
vem
ents
wer
e m
ade
in t
he
syst
em i
den
tifi
cati
on
ro
uti
ne.
A c
om
par
iso
n o
f cla
ssic
al u
nst
ead
y ae
rod
ynam
ic s
olu
tio
ns
was
per
form
ed. T
he
AR
MA
fo
rm r
emai
ns
the
pre
ferr
ed
syst
em re
pre
sen
tati
on
.
A p
aral
lel t
rain
ing
met
ho
d w
as d
evel
op
ed. P
aral
lel t
rain
ing
al
low
s fo
r d
eco
up
led
exc
itat
ion
an
d y
ield
s b
ette
r sy
stem
m
od
els.
An
exc
itat
ion
sig
nal
su
rvey
was
co
nd
uct
ed. T
he
DC
-Ch
irp
g
ave
the
mo
st
con
sist
ent
resu
lts.
H
igh
fr
equ
ency
lim
itat
ion
s o
f th
e cu
rren
t si
gn
als
wer
e id
enti
fied
an
d
eval
uat
ed.
Mo
del
q
ual
ity
"Eva
luat
ion
cr
iter
ia"
wer
e te
sted
. Th
e co
up
led
sta
bili
ty p
red
icti
on
eva
luat
ion
ap
pea
rs to
giv
e th
e b
est
"mo
del
qu
alit
y" in
dic
atio
n.
