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A NEW APPROACH TO FINITE STATE MODELLING OF
7v
C . ~ e n k a t e s a n ' and P .P. Friedmann
Mechanical, Aerospace and Nuclear E n g i n e e r i n g Department
U n i v e r s i t y of C a l i f o r n i a , Los Angeles
Los Angeles, C a l i f o r n i a 90024
Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1986-865
wb
Ub
Abstract
reduced f r e q u e n c y , k = - o r
Qr
This paper p r e s e n t s a novel technique f o r f o r mulating a high q u a l i t y f i n i t e s t a t e unsteady
aerodynamic model by a p p l y i n g Bode p l o t methods,
used i n c o n t r o l e n g i n e e r i n g . I n d i c i a 1 r e s p o n s e
f u n c t i o n s f o r b o t h f i x e d wing and r o t a r y wing
a p p l i c a t i o n s a r e obtained using these f i n i t e s t a t e
I t i s shown t h a t t h e
u n s t e a d y aerodynamic models.
r o t a r y wing i n d i c i a l r e s p o n s e f u n c t i o n h a s a fundam e n t a l l y d i f f e r e n t c h a r a c t e r i s t i c when compared
t o f i x e d wing i n d i c i a l r e s p o n s e . The r o t a r y wing
i n d i c i a l response function is o s c i l l a t o r y i n
n a t u r e w h i l e t h e f i x e d wing i n d i c i a l r e s p o n s e
f u n c t i o n i s n o n o s c i l l a t o r y . F u r t h e r m o r e i t should
b e emphasized t h a t t h i s i s t h e f i r s t t h a t a r o t a r y wing i n d i c i a l r e s p o n s e f u n c t i o n h a s been p r e s e n t e d
in the literature.
U
l i f t p e r u n i t span of t h e b l a d e
c i r c u l a t o r y l i f t on t h e a i r f o i l
-
equivalent frequency r a t i o ; n
e
=
W
-
RQ
number of b l a d e s i n a r o t o r o r 3 1 4 chord downwash v e l o c i t y
r a d i a l s t a t i o n f o r the typical blade
r
section, r = e
bQ
rotor radius
Laplace v a r i a b l e
nondimensional L a p l a c e v a r i a b l e s ;
-s = - b
s
s s= - Q
- -, s = b- s
R0.75R '
U
Nomenclature
l i f t curve slope
f r e e stream v e l o c i t y
c o e f f i c i e n t s i n a p p r o x i m a t e aerodynamic
transfer function
induced v e l o c i t y normal t o t h e r o t o r
b l a d e semichord
frequency
Theodorsen's l i f t deficiency function
r o t o r R.P.M.
g e n e r a l i z e d T h e o d o r s e n ' s l i f t deficiency function
d e n s i t y of a i r
nondimensional t i m e ;
'I =
0.75R
b
Loewy ' s l i f t d e f i c i e n c y f u n c t i o n
f i x e d wing i n d i c i a l r e s p o n s e f u n c t i o n
thrust coefficient
r o t a r y wing i n d i c i a l r e s p o n s e f u n c t i o n
r e a l and imaginary p a r t s of Theodorsen's l i f t deficiency function
C = F + i G
damping f a c t o r s
1.
r e a l and imaginary p a r t s of Loewy's
l i f t deficiency function C' = F'
iG'
+
transfer function
Hankel f u n c t i o n s of second k i n d of
order n;
H:
=
J~
-
iyn
2TrU
e q u i v a l e n t wake s p a c i n g ;
fi e
=
QQb
B e s s e l f u n c t i o n s of f i r s t k i n d of
order n
m o d i f i e d B e s s e l f u n c t i o n s of second
k i n d of o r d e r n
*
'
Introduction
I t i s w e l l known t h a t t h e r o l e of u n s t e a d y
aerodynamics i s i m p o r t a n t f o r a e r o e l a s t i c s t a b i l i t y
and r e s p o n s e c a l c u l a t i o n s i n b o t h r o t a r y wing and
f i x e d wing a p p l i c a t i o n s . A wide a r r a y of mathem a t i c a l models have been developed t o r e p r e s e n t
u n s t e a d y aerodynamic l o a d s , s t a r t i n g from s i m p l e
and c o m p u t a t i o n a l l y e f f i c i e n t models and c u l m i n a t i n g
a very complicated, computationally expensive
models which a r e c a p a b l e of c a p t u r i n g t h e i n t r i c a t e
d e t a i l s of u n s t e a d y f l o w . Two d i m e n s i o n a l u n s t e a d y
aerodynamic t h e o r i e s , which p r o v i d e a n a l y t i c exp r e s s i o n s f o r t h e u n s t e a d y l o a d s on a moving a i r f o i l , a r e u s u a l l y b a s e d on t h e a s s u m p t i o n of
s i m p l e harmonic motion of t h e a i r f o i l . R e p r e s e n t a t i v e t h e o r i e s i n t h i s category are:
T h i s s t u d y was s u p p o r t e d by NASA Ames R e s e a r c h C e n t e r u n d e r Grant NASA NAG 2-209
A s s i s t a n t Research Engineer
"Professor
of E n g i n e e r i n g and A p p l i e d S c i e n c e , A s s o c i a t e F e l l o w A I M
Released to AIAA to publish in all forms.
( a ) ~ h e o d o r s e n ' s l i n c o m p r e s s i b l e two-dimensional
u n s t e a d y aerodynamic t h e o r y and G r e e n b e r g ' s 2 ext e n s i o n of T h e o d o r s e n ' s t h e o r y which a c c o u n t s f o r
p u l s a t i n g oncoming f l o w v e l o c i t y and c o n s t a n t
a n g l e of a t t a c k ; t h e s e t h e o r i e s have been developed
f o r f i x e d wings; however t h e y have been f r e q u e n t 1
u s e d i n rotary-wing a p p l i c a t i o n s a n d , (b) Loewy ' s
t h e o r y and Shipman and wood's4 t h e o r y , which a r e
a p p l i c a b l e t o a h e l i c o p t e r r o t o r i n hover and
forward f l i g h t , r e s p e c t i v e l y .
3
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The fundamental d i f f e r e n c e between t h e f i x e d
wing u n s t e a d y aerodynamic t h e o r i e s and t h e r o t a r y
wing u n s t e a d y aerodynamic t h e o r i e s l i e s i n t h e
m o d e l l i n g of u n s t e a d y wake d u e t o t h e a i r f o i l
motion. T h e o d o r s e n ' s t h e o r y , f o r t h e f i x e d wing
c a s e s assumes a p l a n a r wake behind t h e a i r f o i l
e x t e n d i n g t o i n f i n i t y , whereas Loewy's t h e o r y , f o r
r o t a r y wings, assumes a n u n s t e a d y wake behind and
beneath t h e r e f e r e n c e a i r f o i l , extending t o inf i n i t y , i n b o t h d i r e c t i o n s ( F i g . 1 ) . These t h e o r i e s h a v e a s i g n i f i c a n t d e f i c i e n c y when a p p l y i n g
them t o a e r o e l a s t i c s t a b i l i t y c a l c u l a t i o n s , s i n c e
t h e a s s u m p t i o n of s i m p l e harmonic motion, upon
which t h e y a r e b a s e d , i m p l i e s t h a t t h e y a r e
s t r i c t l y v a l i d o n l y a t t h e s t a b i l i t y boundary, and
t h u s t h e y p r o v i d e no i n f o r m a t i o n on system damping
before o r a f t e r t h e f l u t t e r condition i s reached.
Thus a s t a n d a r d s t a b i l i t y a n a l y s i s , such a s t h e
r o o t l o c u s method, c a n n o t b e u s e d i n c o n j u n c t i o n
with these t h e o r i e s . Furthermore, t h e s e unsteady
aerodynamic t h e o r i e s a r e n o t s u i t a b l e f o r t h e
a n a l y s i s of a e r o e l a s t i c s y s t e m s w i t h a c t i v e cont r o l s , such a s h i g h e r harmonic c o n t r o l d e v i c e s ,
and t h e t r a n s i e n t r e s p o n s e a n a l y s i s of a e r o e l a s t i c
systems, such a s r o t o r b l a d e r e s p o n s e i n forward
f l i g h t . Thus, t h e r e i s a need f o r u n s t e a d y a e r o dynamic t h e o r i e s which a r e c a p a b l e of m o d e l l i n g
t h e u n s t e a d y aerodynamic l o a d s , i n t h e t i m e domain
f o r f i n i t e t i m e a r b i t r a r y motion of a n a i r f o i l ,
r e p r e s e n t i n g t h e c r o s s - s e c t i o n of a n o s c i l l a t i n g
helicopter rotor blade.
I n t h i s p a p e r t h e term
a r b i t r a r y motion i s u s e d t o d e n o t e growing o r
decaying o s c i l l a t i o n s w i t h a c e r t a i n f r e q u e n c y .
Such a n u n s t e a d y aerodynamic t h e o r y , f o r f i n i t e
t i m e a r b i t r a r y motion of t h e a i r f o i l h a s been
developed r e c e n t l y f o r f i x e d wing a p p l i c a t i o n s .
However, t h e s u c c e s s i n d e v e l o p i n g s i m i l a r t h e o r i e s which a r e s u i t a b l e f o r r o t a r y wing a p p l i c a t i o n s
h a s been l i m i t e d .
-
A b r i e f summary of t h e developments i n f i n i t e
t i m e u n s t e a d y aerodynamic m o d e l l i n g f o r f i x e d wing
a p p l i c a t i o n s i s presented f o r convenience. I n t h e
e a r l y 1 9 4 0 ' s L a p l a c e t r a n s f o r m methods were a p p l i e d
t o t h e problem of u n s t e a d y aerodynamics f o r f i n i t e
t i m e a r b i t r a r y motion of a n a i r f o i l 5 . 6 3 7 .
However,
c e r t a i n c o m p u t a t i o n a l d i f f i c u l t i e s were e n c o u n t e r e d
i n e x t e n d i n g t h e s e t h e o r i e s t o v a l u e s of t h e
L a p l a c e t r a n s f o r m v a r i a b l e a l o n g t h e imaginary
a x i s , which r e p r e s e n t s s i m p l e harmonic motion.
R e c e n t l y , ~ d w a r d sr ~
esolved t h e computational
d i f f i c u l t i e s e n c o u n t e r e d i n t h e a p p l i c a t i o n of
Laplsce transform technique t o t h e unsteady aerodynamics of a two-dimensional a i r f o i l o s c i l l a t i n g
i n i n c o m p r e s s i b l e f l o w . He showed t h a t t h e L a p l a c e
t r a n s f o r m of t h e c i r c u l a t o r y l o a d on t h e a i r f o i l ,
executing a r b i t r a r y motion, i s r e l a t e d t o t h e
p r o d u c t of t h e L a p l a c e t r a n s f o r m of t h e g e n e r a l i z e d
Theodorsen f u n c t i o n c(;) and t h e L a p l a c e t r a n s f o r m
of t h e 314 chord downwash v e l o c i t y of t h e a i r f o i l .
The g e n e r a l i z e d Theodorsen l i f t d e f i c i e n c y
f u n c t i o n ~ ( s ) ,i n t h e L a p l a c e domain, i s t h e same
a s replacing, i k , i n Theodorsen's l i f t deficiency
f u n c t i o n C (k) by t h e nondimensional L a p l a c e t r a n s form v a r i a b l e
;.
Recently, vepa9 derived a n approximate funct i o n a l form f o r t h e aerodynamic t r a n s f e r f u n c t i o n
corresponding t o Theodorsen's l i f t deficiency
f u n c t i o n i n t e r m s of Pade a p p r o x i m a n t s . Knowing
t h e e x a c t l i f t d e f i c i e n c y f u n c t i o n C(k) , v e p a 9
r e p r e s e n t e d C(k) by t h e r a t i o of p o l y n o m i a l s N ( i k ) /
D ( i k ) , where N(ik) and D ( i k ) a r e e q u a l d e g r e e
p o l y n o m i a l s . Vepa e v a l u a t e d t h e c o e f f i c i e n t s of
t h e v a r i o u s t e r m s of t h e p o l y n o m i a l s by a l e a s t
s q u a r e t e c h n i q u e and o b t a i n e d a v e r y good a p p r o x i m a t i o n t o C ( k ) . ~ o w e l l l ' proposed a d i f f e r e n t
procedure f o r obtaining t h e approximate expression
f o r C(k). Dowell's technique u s e s a parameter
i d e n t i f i c a t i o n t e c h n i q u e , i n which t h e t i m e h i s t o r y
of t h e aerodynamic l o a d on t h e a i r f o i l was assumed
t o c o n s i s t of sums of e x p o n e n t i a l s . Applying t h e
fundamental c o r r e s p o n d e n c e between t h e f r e q u e n c y
domain and t i m e domain aerodynamic l o a d s , Dowel1
e v a l u a t e d t h e t i m e c o n s t a n t s and t h e c o e f f i c i e n t s
which p r o v i d e t h e b e s t f i t t o t h e f r e q u e n c y domain
r e p r e s e n t a t i o n of t h e aerodynamic f o r c e s . The
a p p r o x i m a t e r e p r e s e n t a t i o n s of T h e o d o r s e n ' s l i f t
d e f i c i e n c y f u n c t i o n C ( k ) , o b t a i n e d by Vepa and
Dowell, c a n b e u s e d t o model t h e u n s t e a d y aerodynamic l o a d s produced by c o m p l e t e l y a r b i t r a r y , s m a l l ,
t i m e dependent m o t i o n s of a n a i r f o i l .
Since these
a p p r o x i m a t e t r a n s f e r f u n c t i o n s f o r C(k) a r e f i n i t e
degree polynomials, they a r e a l s o r e f e r r e d t o a s
f i n i t e s t a t e models f o r t h e u n s t e a d y aerodynamics.
A b r i e f summary of rotary-wing t y p e u n s t e a d y
aerodynamic t h e o r i e s i s p r o v i d e d n e x t . ~ o e w
was~ ~
t h e f i r s t t o p r e s e n t a c l o s e d form s o l u t i o n f o r t h e
u n s t e a d y aerodynamic l o a d s a c t i n g on t h e c r o s s
s e c t i o n of a r o t o r b l a d e performing s i m p l e harmonic
o s c i l l a t i o n s i n i n c o m p r e s s i b l e f l o w . The wake
model assumed by Loewy i s shown i n F i g u r e 1.
Loewy's u n s t e a d y aerodynamic t h e o r y i s t h e r o t a r y wing c o u n t e r p a r t of T h e o d o r s e n ' s t h e o r y f o r f i x e d
wings. A c l e a r d e s c r i p t i o n of Loewy's t h e o r y c a n
b e found i n Ref. 11. The f i r s t a t t e m p t t o g e n e r a l i z e Loewy's t h e o r y f o r a r b i t r a r y m o t i o n s was undert a k e n by D i n y a v a r i and Friedmann12. I n Ref. 1 2 ,
Loewy ' s l i f t d e f i c i e n c y f u n c t i o n C ' was g e n e r a l i z e d
by r e p l a c i n g , i k , i n t h e l i f t d e f i c i e n c y f u n c t i o n
by t h e nondimensional L a p l a c e o p e r a t o r
Followi n g ~ o w e l l ' s l Op r o c e d u r e , a f i n i t e s t a t e Pade
a p p r o x i m a t i o n t o Loewy's l i f t d e f i c i e n c y was obt a i n e d . However, t h e a p p r o x i m a t e model o b t a i n e d
i n t h i s f a s h i o n was i n c a p a b l e of c a p t u r i n g t h e
o s c i l l a t o r y b e h a v i o r of t h e Loewy's l i f t d e f i c i e n c y
f u n c t i o n 1 2 . T h i s was a s e r i o u s shortcomming
b e c a u s e t h e o s c i l l a t o r y n a t u r e of Loewy's l i f t
deficiency function is a unique c h a r a c t e r i s t i c
of t h e u n s t e a d y aerodynamic l o a d s a c t i n g on r o t a t ing blades.
s.
The p r i m a r y o b j e c t i v e s of t h i s p a p e r a r e :
(a) t o present a novel technique f o r formulating
a h i g h q u a l i t y f i n i t e s t a t e u n s t e a d y aerodynamic
model s u i t a b l e f o r b o t h f i x e d wing and rotary-wing
a p p l i c a t i o n s , by u s i n g Bode p l o t methods employed
i n c o n t r o l t h e o r y , (b) a p p l y t h i s t e c h n i q u e t o
t h e f o r m u l a t i o n of t h e u n s t e a d y aerodynamic t r a n s f e r f u n c t i o n f o r T h e o d o r s e n ' s t h e o r y and a r o t a r y wing u n s t e a d y t h e o r y , such a s Loewy's t h e o r y , f o r
a n a i r f o i l u n d e r g o i n g a r b i t r a r y motion i n t i m e
domain a n d , ( c ) u s e t h e t e c h n i q u e t o f o r m u l a t e a
rotary-wing t y p e i n d i c i a 1 r e s p o n s e f u n c t i o n and
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compare i t t o i t s f i x e d wing c o u n t e r p a r t .
F i r s t a n o v e l t e c h n i q u e f o r i d e n t i f y i n g and
f o r m u l a t i n g t h e m a t h e m a t i c a l f orm of t h e f i n i t e
s t a t e u n s t e a d y aerodynamic t r a n s f e r f u n c t i o n i s
presented.
T h i s t e c h n i q u e i s based upon t h e Bode
P l o t method13, u s e d i n c o n t r o l s y s t e m s e n g i n e e r i n g .
S u b s e q u e n t l y t h e f i n i t e s t a t e u n s t e a d y aerodynamic
models a r e u s e d t o o b t a i n i n d i c i a l r e s p o n s e
f u n c t i o n s f o r b o t h a r o t a r y wing and f i x e d wing
applications.
I t i s shown t h a t t h e r o t a r y wing
i n d i c i a l response function is q u a l i t a t i v e l y
d i f f e r e n t i n n a t u r e when compared t o i t s f i x e d
wing c o u n t e r p a r t . F o r f i x e d wings, t h e i n d i c i a l
response function e x h i b i t s a steady exponential
d e c a y i n g form whereas f o r r o t a r y wings t h e i n d i c i a l
r e s p o n s e f u n c t i o n h a s a n o s c i l l a t o r y form w i t h
e x p o n e n t i a l decay. T h e r e f o r e , t h e r o t a r y wing
i n d i c i a l response overshoots t h e steady value a t
c e r t a i n s p e c i f i c t i m e s . T h i s f e a t u r e of t h e
r o t a r y wing i n d i c i a l r e s p o n s e f u n c t i o n h a s been
a l s o observed e x p e r i m e n t a l l y 1 4 .
2.
2.1
F i n i t e S t a t e Modelling of Unsteady Aerodynamics
G e n e r a l Overview
I n t h e d e r i v a t i o n of u n s t e a d y aerodynamic
l o a d on a two-dimensional a i r f o i l i n incompressi b l e flow, f o r f i n i t e time a r b i t r a r y motion,
~ d w a r d sh~a s shown t h a t t h e g e n e r a l i z e d T h e o d o r s e n ' s
l i f t d e f i c i e n c y f u n c t i o n ~ ( s ) ,o b t a i n e d f o r
a r b i t r a r y motion of t h e a i r f o i l , c a n b e e x p r e s s e d
as
K1
c(S) =
(1)
K ~ ( " + Kl ( 3
(a
m
1
by minimizing t h e e r r o r
i=l
L
- A
where i = 1
m r e f e r t o t h e v a l u e of k a t s e l e c t ed m p o i n t s . Using a f o u r t h d e g r e e p o l y n o m i a l s f o r
N and D ; i . e . , a [ 4 , 4 ] Pade a p p r o x i m a t i o n . Vepa
e v a l u a t e d t h e c o e f f i c i e n t s a ' s and b ' s , by minimiz-
.. .
-
m
.
Z
1 [~(k.)~(ik.)-~(ik,)]
j=1
J
J
The [ 4 , 4 ] Pade approximant o b t a i n e d f o r t h i s c a s e
was found t o b e a v e r y good a p p r o x i m a t i o n t o t h e
l i f t deficiency function C(k).
ing t h e modified f u n c t i o n
~ o w e l l l Ou s e d a d i f f e r e n t a p p r o a c h f o r o b t a i n i n g t h e a p p r o x i m a t e f u n c t i o n a l form of C ( k ) . F i r s t
h e assumed t h e i n d i c i a l r e s p o n s e f u n c t i o n f o r a
two-dimensional a i r f o i l i n i n c o m p r e s s i b l e f l o w .
S i n c e ones^ d e r i v e d a n a p p r o x i m a t e f u n c t i o n a l
form f o r Wagner's i n d i c i a l l i f t f u n c t i o n , g i v e n a s
ow ell" assumed a n i n d i c i a l r e s p o n s e f u n c t i o n ,
s i m i l a r i n form t o Eq. ( 7 ) ,
For a s t a b l e , o r convergent i n d i c i a l response
f u n c t i o n , a l l b . ' s should b e n e g a t i v e .
J
The i n d i c i a l r e s p o n s e f u n c t i o n i s r e l a t e d t o
Theodorsen's l i f t deficiency f u n c t i o n through a
F o u r i e r t r a n s f orm15, i. e . ,
The c i r c u l a t o r y p o r t i o n of t h e l i f t p e r u n i t span
of t h e a i r f o i l i n t h e L a p l a c e domain c a n b e
written a s
Taking t h e i n v e r s e t r a n s f o r m of Eq. ( 9 )
where Q(s) r e p r e s e n t s t h e L a p l a c e t r a n s f o r m of t h e
314-chord downwash v e l o c i t y .
For harmonic a i r f o i l mot i o n , t h e g e n e r a l i z e d
Theodorsen l i f t d e f i c i e n c y f u n c t i o n i s o b t a i n e d
by i k i n Eq. ( I ) , i . e . ,
by r e p l a c i n g
s
S i n c e C(k) g i v e n i n Eq. ( 3 ) i s known e x a c t l y ,
v e p a 9 c o n s t r u c t e d a Pade approximant f o r C ( k ) , by
assuming
C(k) = F
where N ( i k
degrees.
+
i G Z-
N(ik)
D(ik)
and D ( i k ) a r e p o l y n o m i a l s of equal
For n t h d e g r e e p o l y n o m i a l s , t h e a p p r o x i m a t i o n i s
r e f e r r e d t o a s a n [ n , n ] Pade a p p r o x i m a n t . The
a n and b l
bn c a n b e e v a l u a t e d
coefficients a1
..
..
S u b s t i t u t i n g Eq.
C(k) becomes
(8) i n Eq. (10) and i n t e g r a t i n g ,
I n Ref. 1 0 , Dowel1 f i r s t s e l e c t e d t h e b j ' s t o
b e on t h e n e g a t i v e r e a l a x i s , i - e . , n e g a t i v e v a l u e s
f o r a l l b j ' s , t o a c c o u n t f o r a c o n v e r g e n t QF.w.(.r)
g i v e n by Eq. ( 8 ) . S u b s e q u e n t l y , t h e c o e f f i c i e n t
a j ' s were d e t e r m i n e d . Using t h i s p r o c e d u r e Dowel1
o b t a i n e d a v e r y good a p p r o x i m a t i o n t o T h e o d o r s e n ' s
l i f t d e f i c i e n c y f u n c t i o n C(k)
The main d i f f e r e n c e
between V e p a ' s p r o c e d u r e and Dowell's p r o c e d u r e i s
t h a t Vepa d i r e c t l y assumed a n a p p r o x i m a t e form f o r
C(k) i n t e r m s of a r a t i o of p o l y n o m i a l s , whereas
Dowel1 s t a r t e d by assuming t h e i n d i c i a l r e s p o n s e
f u n c t i o n and d e t e r m i n e d t h e c o e f f i c i e n t s a j ' s , a f t e r
a s s i g n i n g s p e c i f i c v a l u e s t o a l l b j ' s . It s h o u l d
b e n o t e d t h a t Dowell's p r o c e d u r e i s based on t h e
a s s u m p t i o n t h a t t h e f u n c t i o n a l form of t h e i n d i c i a l
r e s p o n s e f u n c t i o n i s known a p r i o r i .
.
I n R e f . 1 2 , D i n y a v a r i and Friedmann used
Dowell's procedure t o o b t a i n t h e approximate transf e r f u n c t i o n f o r Loewy ' s l i f t d e f i c i e n c y f u n c t i o n ,
f o r t h e c r o s s - s e c t i o n of a h e l i c o p t e r b l a d e .
The
p o l e s of t h e t r a n s f e r f u n c t i o n were p l a c e d on t h e
n e g a t i v e r e a l a x i s , i n a manner c o n s i s t e n t w i t h
ow ell's a s s u m p t i o n of n e g a t i v e v a l u e s f o r a l l b j 's.
And t h e c o e f f i c i e n t s a j ' s were d e t e r m i n e d . The
approximate t r a n s f e r f u n c t i o n obtained i n t h i s
manner f a i l e d t o c a p t u r e t h e o s c i l l a t o r y b e h a v i o r
of Loewy's l i f t d e f i c i e n c y f u n c t i o n and t h u s c o r r e l a t i o n between t h e a p p r o x i m a t e model and t h e
e x a c t v a l u e of Loewy's f u n c t i o n was found t o b e
poor.
I t i s shown below t h a t t h e Bode p l o t of t h e
aerodynamic t r a n s f e r f u n c t i o n p r o v i d e s t h e necess a r y i n f o r m a t i o n r e g a r d i n g t h e n a t u r e of t h e p o l e s
and z e r o s ( s u c h a s r e a l o r complex) a s w e l l a s
t h e i r l o c a t i o n s i n t h e Laplace plane. This information c a n b e used t o f o r m u l a t e t h e a p p r o x i m a t e
aerodynamic t r a n s f e r f u n c t i o n s i n a n a c c u r a t e and
r e l i a b l e manner.
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2.2
B r i e f D e s c r i p t i o n of t h e Bode P l o t and I t s
R o l e i n Unsteady Aerodynamic Modeling
One of t h e methods f r e q u e n t l y u s e d i n t h e
a n a l y s i s and d e s i g n of c o n t r o l systems i s t h e f r e quency r e s p o n s e method. When u s i n g t h i s method,
t h e f r e q u e n c y of t h e i n p u t i s v a r i e d o v e r a wide
r a n g e and t h e r e s u l t i n g o u t p u t r e s p o n s e i s s t u d i e d .
Using t h i s method, t h e t r a n s f e r f u n c t i o n s of comp l i c a t e d systems c a n b e d e t e r m i n e d . T h i s p a r t i c u l a r a s p e c t of d e t e r m i n i n g t h e t r a n s f e r f u n c t i o n
from t h e f r e q u e n c y r e s p o n s e c u r v e c a n a l s o b e u s e d
t o formulate approximate t r a n s f e r f u n c t i o n s f o r
t h e u n s t e a d y aerodynamics of a two-dimensional
a i r f o i l o s c i l l a t i n g i n incompressible flow.
Furthermore, t h e method i s e q u a l l y a p p l i c a b l e t o
b o t h f i x e d wing t y p e o r r o t a r y wing t y p e of unsteady
aerodynamic t h e o r i e s where t h e l i f t d e f i c i e n c y
f u n c t i o n p l a y s e s s e n t i a l l y t h e r o l e of a t r a n s f e r
f u n c t i o n , which r e l a t e s t h e 314-chord downwash
v e l o c i t y t o t h e aerodynamic l o a d a c t i n g on t h e
airfoil.
Since the l i f t deficiency functions,
s u c h a s T h e o d o r s e n ' s f o r t h e f i x e d wing c a s e and
Loewy's f o r t h e r o t a r y wing c a s e a r e known e x a c t l y
f o r s i m p l e harmonic motion of t h e a i r f o i l , t h e
approximate t r a n s f e r f u n c t i o n s can be formulated
by t h e a p p l i c a t i o n of f r e q u e n c y - r e s p o n s e t e c h n i q u e s
used i n c o n t r o l system e n g i n e e r i n g .
One of t h e methods used f o r r e p r e s e n t i n g t h e
f r e q u e n c y - r e s p o n s e of a t r a n s f e r f u n c t i o n i s t h e
Bode diagram13. The Bode diagram c o n s i s t s of two
p a r t s . One i s a p l o t of t h e l o g a r i t h m of t h e
magnitude of t h e s i n u s o i d a l t r a n s f e r f u n c t i o n and
t h e o t h e r i s a p l o t of t h e p h a s e a n g l e . Both a r e
p l o t t e d a s a f u n c t i o n of f r e q u e n c y on a l o g a r i t h m i c
s c a l e . S i n c e o n l y t h e p l o t of t h e magnitude v e r s u s
t h e f r e q u e n c y i s needed f o r t h e p r e s e n t a p p l i c a t i o n ,
t h e d i s c u s s i o n , g i v e n below, i s r e s t r i c t e d t o t h i s
p a r t i c u l a r a s p e c t of t h e Bode p l o t .
-
The s t a n d a r d r e p r e s e n t a t i o n of t h e Bode p l o t
c o n s i s t s of a l o g a r i t h m i c magnitude of t h e t r a n s f e r
f u n c t i o n G(iw) i . e . , 20 l o g IG(iw)
versus the
f r e q u e n c y w.
I,
S i n c e o u r aim i s t o f o r m u l a t e a n a p p r o x i m a t e
t r a n s f e r function t o t h e unsteady l i f t deficiency
f u n c t i o n we s e e k t h e i n f o r m a t i o n a b o u t t h e q u a l i t a t i v e n a t u r e of t h e p o l e s and z e r o s , i . e . ,
whether t h e y a r e r e a l o r complex and where t h e y a r e
l o c a t e d . T h i s i n f o r m a t i o n can b e o b t a i n e d by
a n a l y z i n g t h e Bode p l o t of t h e l i f t d e f i c i e n c y
f u n c t i o n . A b r i e f d e s c r i p t i o n of t h e c e r t a i n
a s p e c t s of r e a l and complex p o l e s o r z e r o s on t h e
Bode p l o t , which i s c r u c i a l f o r t h e u n d e r s t a n d i n g
of t h i s p a p e r i s p r e s e n t e d below.
Consider a simple t r a n s f e r f u n c t i o n G ( h )
one r e a l p o l e and o n e r e a l z e r o , g i v e n by
with
where a 1 and b l r e p r e s e n t t h e z e r o and p o l e of t h e
t r a n s f e r f u n c t i o n r e s p e c t i v e l y . Note t h a t a 1 and
b l a r e r e a l numbers.
Taking t h e l o g a r i t h m of
IG(iw)
I
F o r t h e s a k e of t h i s d i s c u s s i o n l e t a l = l o b l .
The a s y m p t o t i c b e h a v i o r of t h e f i r s t term i n Eq.
(12), f o r o < < bl, is
-20 l o g
10
dl
+
1 -20 l o g l O l = 0 db
Thus t h e log-magnitude of t h e f i r s t t e r m a t low
f r e q u e n c i e s i s a c o n s t a n t w i t h a v a l u e of 0 db and
t h e low f r e q u e n c y a s y m p t o t e i s a 0 db l i n e , a s
shown i n F i g . 2 . F o r h i g h f r e q u e n c i e s w > > b l ,
t h e f i r s t t e r m of Eq. (12) becomes
Thus a t h i g h f r e q u e n c y , t h e log-magnitude h a s a
-20 d b l d e c a d e s l o p e w i t h r e s p e c t t o loglO(w/bl).
S i m i l a r l y , f o r t h e second t e r m i n Eq. ( 1 3 ) , t h e
low f r e q u e n c y a s y m p t o t e , i . e . , f o r w < < a l , i s
t h e 0 db l i n e and t h e h i g h f r e q u e n c y a s y m p t o t e h a s
a s l o p e of +20 d b l d e c a d e .
Combining t h e a s y m p t o t e s of t h e two t e r m s i n
Eq. (13) t h e a s y m p t o t i c b e h a v i o r of t h e t r a n s f e r
f u n c t i o n l ~ ( i w ) I i s shown i n F i g . 2 .
I t can b e
s e e n t h a t t h e e x a c t c u r v e h a s a smooth change i n
t h e s l o p e , whereas t h e a s y m p t o t i c c u r v e h a s a b r u p t
changes i n s l o p e s . The p o i n t s A and B , a t which
t h e s l o p e of t h e a s y m p t o t e c h a n g e s , a r e c a l l e d
c o r n e r f r e q u e n c i e s . A t p o i n t A , t h e s l o p e changes
from 0 d b l d e c a d e t o -20 d b l d e c a d e i n d i c a t i n g t h e
p r e s e n c e of a r e a l p o l e i s t h e t r a n s f e r f u n c t i o n .
A t p o i n t B, t h e s l o p e changes from -20 d b l d e c a d e
t o 0 d b l d e c a d e , d u e t o t h e p r e s e n c e of a r e a l z e r o
i n the transfer function.
I n F i g . 2, i t c a n b e
s e e n t h a t t h e a s y m p t o t i c b e h a v i o r of t h e e x a c t
c u r v e a t v e r y low and v e r y h i g h f r e q u e n c i e s shows
e q u a l s l o p e s w i t h a magnitude of 0 d b l d e c a d e . The
i m p o r t a n t c o n c l u s i o n t h a t c a n b e drawn from t h i s
o b s e r v a t i o n i s t h a t whenever t h e s l o p e s of t h e
a s y m p t o t e s a t v e r y low and v e r y h i g h f r e q u e n c i e s
a r e t h e same, t h e n t h e t r a n s f e r f u n c t i o n s h o u l d
h a v e e q u a l number of p o l e s and z e r o s . T h i s conc l u s i o n w i l l b e u s e f u l i n determining t h e qualit a t i v e n a t u r e of t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n
t o b e formulated f o r t h e l i f t deficiency function.
The i n f l u e n c e of complex p o l e s and complex
z e r o s , on t h e Bode p l o t of t h e t r a n s f e r i s des c r i b e d n e x t . L e t u s assume a t r a n s f e r f u n c t i o n
of t h e form,
2.3
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where C1 and C2 r e p r e s e n t t h e damping f a c t o r s .
When 51 and C 2 a r e g r e a t e r t h a n 1 . 0 , t h e quadr a t i c e x p r e s s i o n i n t h e numerator and demoninator
c a n b e w r i t t e n a s p r o d u c t s of two f i r s t o r d e r t e r m s
w i t h r e a l z e r o s and r e a l p o l e s , and t h e t e c h n i q u e
of p l o t t i n g t h e a s y m p t o t e s f o l l o w s t h e same procedure outlined previously f o r t h e case with s i n g l e
p o l e and s i n g l e z e r o . When 51 and 52 a r e l e s s t h a n
1 . 0 , t h e q u a d r a t i c t e r m s i n t h e numerator and denominator of Eq. (14) w i l l have two complex conjugate roots.
The a s y m p t o t i c b e h a v i o r of t h e
t r a n s f e r f u n c t i o n , g i v e n i n Eq. (14) i s shown i n
F i g . 3 . I n d e a l i n g w i t h t h e a s y m p t o t i c b e h a v i o r of
t h e t r a n s f e r f u n c t i o n i t i s i m p o r t a n t t o mention
t h a t t h e a s y m p t o t e s a r e i n d e p e n d e n t of t h e v a l u e s
of t h e damping f a c t o r s 51 and 52. A c t u a l l y , t h e
d e r i v a t i o n of t h e e x a c t c u r v e from t h e a s y m p t o t e s
i s governed by t h e v a l u e s of 51 and 52. I n F i g . 3 ,
t h e e x a c t c u r v e s a r e p l o t t e d f o r two s e t s of v a l u e s
of 51 and 52. For t h e f i r s t c a s e , 51 = 52 = 0 . 1 ,
t h e e x a c t c u r v e shows peaks n e a r t h e c o r n e r f r e q u e n c i e s l o c a t e d a t A and B . Using Eq. ( 1 4 ) , w i t h
52 = 52 = 0.10 o n e f i n d s t h a t t h e t r a n s f e r f u n c t i o n
h a s complex p o l e s and z e r o s . T h e r e f o r e b a s e d on
t h e c h a r a c t e r i s t i c s of t h e e x a c t Bode p l o t , of t h e
t r a n s f e r f u n c t i o n , one can c o n c l u d e t h a t t h e p r e s e n c e of p e a k s i n t h e Bode p l o t a r e i n d i c a t i v e of
t h e e x i s t e n c e of complex p o l e s o r z e r o s i n t h e
t r a n s f e r f u n c t i o n . For t h e second c a s e , when 51 =
52 = 1 . 0 , t h e e x a c t c u r v e d o e s n o t e x h i b i t any
peak ( F i g . 1 ) . T h i s c a s e c o r r e s p o n d s t o two e q u a l
r e a l p o l e s and two e q u a l r e a l z e r o s i n t h e t r a n s f e r f u n c t i o n . As a r e s u l t , t h e s l o p e change i n t h e
a s y m p t o t e i s -40 d b l d e c a d e f o r two e q u a l r e a l p o l e s
and +40 d b l d e c a d e f o r two e q u a l r e a l z e r o s .
The i m p o r t a n t c o n c l u s i o n s which c a n b e g l e a n e d
from t h i s d i s c u s s i o n of t h e p r o p e r t i e s of t h e Bode
p l o t s a r e summarized below:
( a ) a +20 d b / d e c a d e change i n s l o p e i n t h e
a s y m p t o t e s i s i n d i c a t i v e of t h e p r e s e n c e of a r e a l
z e r o and a -20 d b l d e c a d e change i n s l o p e i n d i c a t e s
t h e p r e s e n c e of a r e a l p o l e i n t h e t r a n s f e r
function;
( b ) peaks i n t h e Bode p l o t i n d i c a t e t h e p r e s e n c e of complex p o l e s and complex z e r o s i n t h e
t r a n s f e r function;
(c) a -40 d b l d e c a d e change i n s l o p e i n d i c a t e s
t h e p r e s e n c e of e i t h e r complex z e r o s o r two e q u a l
r e a l zeros;
( d ) whenever t h e s l o p e s of t h e a s y m p t o t e s of
t h e t r a n s f e r f u n c t i o n a r e e q u a l , a t low and a t h i g h
frequencies, then t h e t r a n s f e r f u n c t i o n should
h a v e e q u a l number of p o l e s and z e r o s .
These c o n c l u s i o n s a r e d i r e c t l y a p p l i c a b l e t o
t h e f o r m u l a t i o n of a p p r o x i m a t i o n s t o t h e l i f t
d e f i c i e n c y f u n c t i o n which h a s t h e r o l e of a n a e r o dynamic t r a n s f e r f u n c t i o n i n u n s t e a d y aerodynamics.
F i n i t e S t a t e Modelling of T h e o d o r s e n ' s L i f t
Deficiency Function
T h e o d o r s e n ' s l i f t d e f i c i e n c y f u n c t i o n , f o r two
d i m e n s i o n a l a i r f o i l e x e c u t i n g a s i m p l e harmonic
motion i n i n c o m p r e s s i b l e f l o w , i s g i v e n i n e x a c t
form by15
The r e a l and imaginary p a r t s of C(k) a r e shown i n
F i g . 4 . F i g u r e 5 shows t h e Bode p l o t of Theodor..
s e n ' s l i f t d e f i c i e n c y function. It can be seen
from F i g . 5 t h a t t h e low f r e q u e n c y and t h e h i g h
frequency asymptotes have equal s l o p e s with a v a l u e
of 0 d b l d e c a d e . Also, t h e Bode p l o t d o e s n o t show
any p e a k s and t h e s l o p e of t h e e x a c t c u r v e a t any
p o i n t i s l e s s t h a n -20 d b l d e c a d e . T h e r e f o r e , u s i n g
t h e g e n e r a l p r o p e r t i e s of t h e Bode p l o t s p r e s e n t e d
b e f o r e , t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n f o r C(k)
must have e q u a l number of p o l e s and z e r o s . F u r t h e r more, t h e p o l e s and z e r o s s h o u l d b e r e a l . Based
on t h e s e c o n s i d e r a t i o n s , a second d e g r e e Pade
a p p r o x i m a t i o n f o r C ( k ) , i n t h e form of
C(k)
'
0.5(ik
(it
+
al)(ik
+
a2)
+ bl) ( i k + b2)
(16)
c a n b e assumed. The t r a n s f e r f u n c t i o n s a t i s f i e s
t h e c o n d i t i o n t h a t C(k) a p p r o a c h e s 0 . 5 a s k app r o a c h e s i n f i n i t y . Imposing t h e c o n s t a i n t t h a t
C(k) = 1 f o r k = 0 , t h e c o e f f i c i e n t s a l , a 2 , b l , b2
a r e d e t e r m i n e d by a l e a s t s q u a r e t e c h n i q u e , by
e q u a t i n g t h e r e a l and imaginary p a r t s of t h e t r a n s f e r f u n c t i o n w i t h t h e r e a l and imaginary p a r t s of
The a p p r o x i m a t e f u n c t i o n
t h e e x a c t f u n c t i o n C(k)
o b t a i n e d i n t h i s manner i s g i v e n by
.
+
0.651)
0.5(ik + 0.135)(ik
0.0965) ( i k + 0.4555)
C(k) = ( i k
+
(17)
The r e a l and imaginary p a r t s of t h e a p p r o x i m a t e
t r a n s f e r f u n c t i o n a r e shown a l o n g w i t h t h e e x a c t
f u n c t i o n s i n Fig. 4. It can be seen t h a t t h e
a p p r o x i m a t e f u n c t i o n d e v i a t e s s l i g h t l y from t h e
exact function i n t h e frequency range
0.05.
F o r t h e s a k e of comparison, J o n e s
0.05 5 k
a p p r o x i m a t i o n t o Theodorsen's f u n c t i o n C(k) i s a l s o
p l o t t e d i n Fig. 4.
The l o c a t i o n of t h e p o l e s ( P i , P2) and z e r o s
(Z1, 22) and t h e a s y m p t o t i c b e h a v i o r of t h e a p p r o x i mate f u n c t i o n , g i v e n i n Eq. (17) a r e shown i n F i g .
5. At v e r y low f r e q u e n c i e s , t h e s l o p e of t h e
a s y m p t o t e i s 0 d b l d e c a d e and t h e a s y m p t o t e i s t h e
0. db l i n e . A t p o l e P i = -0.0965, t h e s l o p e of t h e
a s y m p t o t e becomes -20 d b l d e c a d e . The t r a n s f e r
f u n c t i o n h a s a z e r o a t Z1 = 0.135 and hence t h e r e i s
a change of s l o p e of +20 d b l d e c a d e . The combined
e f f e c t s of t h e p o l e P I and z e r o , Z1 c a u s e s t h e
s l o p e of t h e a s y m p t o t e a t Z1 t o b e 0 d b l d e c a d e ,
which i s i n d i c a t e d by t h e l i n e ZlP1. A t p o i n t P2,
t h e f u n c t i o n h a s a p o l e P2 = -0.4555 and t h e s l o p e
of t h e a s y m p t o t e becomes -20 d b l d e c a d e . At p o i n t
2 2 , where t h e r e i s a z e r o , 22 = -0.651, t h e s l o p e
of t h e a s y m p t o t e a g a i n r e v e r t s back t o 0 d b l d e c a d e .
By i n c l u d i n g a n a d d i t i o n a l p o l e and z e r o i n
t h e t r a n s f e r f u n c t i o n g i v e n i n Eq. ( 1 6 ) , a t h i r d
d e g r e e [ 3 , 3 ] Pade approximat i o n f o r T h e o d o r s e n ' s
l i f t deficiency f u n c t i o n i s obtained. A f t e r inc l u d i n g t h e e f f e c t of t h e a d d i t i o n a l p o l e and z e r o
t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n i s g i v e n by
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C(k)
0 . 0 8 8 ) ( i k + 0 . 3 7 ) ( i k + 0.922)
' 0 . 5( (i ikk ++0.072)
( i k + 0 . 2 6 1 ) ( i k + 0.80) (18)
The r e a l and imaginary p a r t s of t h e e x a c t C(k)
function along with t h e approximate t r a n s f e r
f u n c t i o n , g i v e n i n Eq. (18) a r e shown i n F i g . 6.
I t i s e v i d e n t t h a t t h e [ 3 , 3 ] Pade a p p r o x i m a t i o n
g i v e s e x c e l l e n t c o r r e l a t i o n w i t h t h e e x a c t C(k)
f u n c t i o n s . The l o c a t i o n of t h e p o l e s and z e r o s
and t h e a s y m p t o t i c b e h a v i o r of t h e a p p r o x i m a t e
t r a n s f e r f u n c t i o n , g i v e n i n Eq. ( 1 8 ) , a r e shown
i n F i g . 7. I t c a n b e s e e n from F i g . 7 and F i g . 5
t h a t t h e p o l e s and z e r o s of t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n l i e i n t h e r a n g e of k where t h e Bode
p l o t of t h e e x a c t t r a n s f e r f u n c t i o n h a s a h i g h e r
s l o p e t h a n a t o t h e r v a l u e s of k , i . e . ,
This p a r t i c u l a r f e a t u r e associated
0.06 5 k 5 1 . O .
w i t h t h e l o c a t i o n of t h e p o l e s and z e r o s i s v e r y
useful f o r estimating t h e i r i n i t i a l values a s
r e q u i r e d when c o n s t r u c t i n g t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n by u s i n g a n o n l i n e a r l e a s t s q u a r e s
technique.
The i n d i c i a l r e s p o n s e f u n c t i o n c a n b e o b t a i n e d
by t a k i n g t h e i n v e r s e F o u r i e r t r a n s f o r m of t h e
a p p r o x i m a t e t r a n s f e r f u n c t i o n , which h a s been
o b t a i n e d above. Using t h e [ 2 , 2 ] Pade a p p r o x i m a t i o n
g i v e n i n Eq. ( 1 7 ) , t h e i n d i c i a l r e s p o n s e f u n c t i o n
is
and behind t h e r e f e r e n c e a i r f o i l . Loewy's t h e o r y
i s i n t e n d e d f o r l i g h t l y l o a d e d r o t o r s ( i . e . , low
i n f l o w c o n d i t i o n s ) and l i k e T h e o d o r s e n ' s t h e o r y
i t i s a l s o b a s e d on t h e a s s u m p t i o n of s i m p l e h a r monic motion of t h e r e f e r e n c e a i r f o i l .
Loewy ' s l i f t d e f i c i e n c y f u n c t i o n , i n f r e q u e n c y
domain, f o r t h e c o l l e c t i v e mode of t h e r o t o r ,
where a l l t h e b l a d e s move i n p h a s e , i s g i v e n b y l 1
where t h e wake w e i g h t i n g f u n c t i o n W i s
- -
The q u a n t i t i e s he,me f o r a t y p i c a l b l a d e s e c t i o n
a t a r a d i a l d i s t a n c e r from t h e a x i s of r o t a t i o n ,
f o r a r o t o r with Q blades, a r e defined a s
ones^ a p p r o x i m a t i o n t o Wagner's i n d i c i a l r e s p o n s e
function is
Comparing t h e s e two i n d i c i a l r e s p o n s e f u n c t i o n s ,
it can b e seen t h a t t h e i n d i c i a l response obtained
i n t h i s s t u d y h a s s m a l l e r t i m e c o n s t a n t when compared t o those i n Jones' approximation.
The i n d i c i a l r e s p o n s e f u n c t i o n o b t a i n e d f o r
t h e [ 3 , 3 ] Pade a p p r o x i m a t i o n , g i v e n by Eq. ( l 8 ) , i s
I n t h e -d e f i n i t i o n of t h e e q u i v a l e n t f r e q u e n c y
r a t i o me, i t s h o u l d b e n o t e d t h a t f o r a g i v e n
b l a d e s e c t i o n -a t a r a d i a l s t a t i o n r from t h e a x i s
of r o t a t i o n , me depends on t h e reduced f r e q u e n c y k .
It w i l l b e shown t h a t t h e t e c h n i q u e based on
t h e Bode p l o t , developed i n t h i s p a p e r , i s s u c c e s s f u l i n formulating t h e approximate t r a n s f e r
f u n c t i o n f o r t h e Loewy's l i f t d e f i c i e n c y f u n c t i o n .
To i l l u s t r a t e t h e method, two examples were s e l e c t ed. The f i r s t example i s t h e same a s t h a t con= 3.0.
s i d e r e d i n R e f . 1 2 , w i t h I?, = 4 . 0 and
This case corresponds t o a blade section located
a t a r a d i a l s t a t i o n 0.8R.
The r o t o r c o n s i s t s of
~ 0.0667.
f o u r b l a d e s w i t h a b l a d e semichord b / =
The r o t o r i s o p e r a t i n g a t a t h r u s t c o e f f i c i e n t
and t h e c o r r e s p o n d i n g i n f l o w r a t i o i s
;,
2.4
F i n i t e S t a t e Modelling of Loewy's L i f t
Deficiency Function
The p r o c e d u r e which h a s been d e s c r i b e d i n t h e
previous s e c t i o n f o r c o n s t r u c t i n g approximations
t o Theodorsen's l i f t d e f i c i e n c y function i s a l s o
a p p l i c a b l e t o t h e f o r m u l a t i o n of a p p r o x i m a t i o n s t o
Loewy's l i f t d e f i c i e n c y f u n c t i o n , which i s t h e
r o t a r y wing c o u n t e r p a r t of Theodorsen's t h e o r y .
Loewy's r o t a r y wing u n s t e a d y a i r f o i l t h e o r y r e p r e s e n t s a n approximation t o t h e unsteady aerodynamic l o a d s a c t i n g on a r o t o r b l a d e c r o s s s e c t i o n
i n h o v e r . The e f f e c t s of t h e s p i r a l r e t u r n i n g
wake b e n e a t h t h e r o t o r , shown i n F i g . 1 , a r e cons i d e r e d i n a n a p p r o x i m a t e manner. These wake
l a y e r s r e p r e s e n t wakes shed by o t h e r b l a d e s , a s
well a s t h e reference blade i n previous revolutions.
The a p p r o x i m a t i o n c o n s i s t s of t h e a s s u m p t i o n t h a t
t h e wake l a y e r s e x t e n d t o i n f i n i t y b e f o r e and
0.17.
These v a l u e s of CT and Xo a r e
unre'asonably h i g h f o r a h e l i c o p t e r . However t h i s
c a s e was s e l e c t e d t o t e s t t h e method f o r a n extreme
c a s e . For t h e second example CT = 0.005 and t h e
b l a d e s e c t i o n was t a k e n a t 0.75R.
The r o t o r cons i s t s of f o u r b l a d e s and t h e b l a d e s e m i c h o d i s
i s b/R
=
0.024.
The i n f l o w r a t i o i s A.
1"
=
12 = 0 '05.
v
-
The c o r r e s p o n d i n g v a l u e s of he and Fe a r e he =
T h i s example i s r e p r e 3.2725 and Ge = 7.8125.
s e n t a t i v e of a t y p i c a l h e l i c o p t e r a p p l i c a t i o n .
The F i r s t Example
The p r e s c r i b e d p a r a m e t e r s f o r t h i s c a s e a r e :
o b t a i n e d by a d d i n g a d d i t i o n a l p o l e s and z e r o s t o
t h e t r a n s f e r f u n c t i o n g i v e n i n Eq. ( 2 4 ) .
a r e i d e n t i c a l t o t h o s e i n Ref. 1 2 .
The r e a l and imaginary p a r t s of t h e Loewy's e x a c t
l i f t d e f i c i e n c y f u n c t i o n C ' a r e shown i n F i g . 8 .
Both t h e r e a l ( F ' ) , and imaginary p a r t s (G'), of
Loewy's l i f t d e f i c i e n c y f u n c t i o n e x h i b i t a n osc i l l a t o r y b e h a v i o r . A l s o shown i n F i g . 8 i s t h e
approximate l i f t d e f i c i e n c y f u n c t i o n obtained i n
Ref. 1 2 , u s i n g a [ 5 , 5 ] Pade approximant. The
agreement between t h i s a p p r o x i m a t i o n and t h e e x a c t
v a l u e s of Loewy's l i f t d e f i c i e n c y f u n c t i o n i s poor.
F i g u r e 9 shows t h e Bode p l o t of t h e Loewy's l i f t
This log-frequency curve
deficiency function C'
e x h i b i t s p e a k s and v a l l e y s . A s mentioned p r e v i o u s l y , t h e p r e s e n c e of peaks and v a l l e y s i n t h e Bode
p l o t i s i n d i c a t i v e of complex p o l e s and complex
zeros i n t h e t r a n s f e r function. Therefore, t h e
a p p r o x i m a t e t r a n s f e r f u n c t i o n f o r C ' must h a v e
complex p o l e s and complex z e r o s . Based on t h i s
requirement, t h e approximate t r a n s f e r f u n c t i o n
s e l e c t e d f o r C ' h a s t h e form
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.
C'(k,rne,he)
-
+
O.5(ik
(ik
+
+
+
a l ) [ ( i ~ +) ~
bl) [ ( i k l 2
ika2
ikb2
+
+
a3] [ (ik12
+
2
b3 1 [ ( i k )
+
+
iLa4
ikb4
+ a 51
+ b5]
An i m p o r t a n t a s p e c t of t h e o s c i l l a t o r y beh a v i o r of Loewy's l i f t d e f i c i e n c y f u n c t i o n i s cons i d e r e d n e x t . From l i n e a r i n c o m p r e s s i b l e , twod i m e n s i o n a l u n s t e a d y aerodynamics t h e o r y f o r f i x e d
wings, one can p r o v e m a t h e m a t i c a l l y t h a t Wagner's
i n d i c i a l r e s p o n s e f u n c t i o n and T h e o d o r s e n ' s l i f t
d e f i c i e n c y f u n c t i o n a r e r e l a t e d by a F o u r i e r t r a n s f o r m l 5 . S i n c e Loewy's t h e o r y i s t h e rotary-wing
c o u n t e r p a r t of T h e o d o r s e n ' s t h e o r y by a n a l o g y i t
i s e v i d e n t t h a t t h e rotary-wing i n d i c i a l r e s p o n s e
f u n c t i o n and Loewy's l i f t d e f i c i e n c y f u n c t i o n c a n
a l s o b e r e l a t e d by a F o u r i e r t r a n s f o r m . Thus t h e
rotary-wing i n d i c i a l r e s p o n s e f u n c t i o n c a n b e obt a i n e d from t h e f o l l o w i n g r e l a t i o n ,
(24)
The a p p r o x i m a t e t r a n s f e r f u n c t i o n g i v e n i n Eq.
( 2 4 ) , s a t i s f i e s t h e h i g h f r e q u e n c y l i m i t , t h u s when
The a p p r o x i k a p p r o a c h e s m, C ' a p p r o a c h e s 0.50.
mate f u n c t i o n g i v e n by Eq. ( 2 4 ) , i s c a p a b l e o f
r e p r e s e n t i n g t h e complex p o l e s and complex z e r o s ,
depending on t h e v a l u e s of t h e c o e f f i c i e n t b 2 , b3,
S i n c e t h e v a l u e s of
bq, b5 and a 2 , a 3 , a q , a:.
t h e e x a c t Loewy l i f t d e f i c i e n c y f u n c t i o n C ' a r e
known a t v a r i o u s reduced f r e q u e n c i e s k , t h e c o e f a 5 and b l
b5 c a n b e e v a l u f i c i e n t a1
ated. I n t h i s case, t h e c o e f f i c i e n t s a r e evaluated
u s i n g a n o n l i n e a r l e a s t s q u a r e t e c h n i q u e b a s e d on
t h e Fletcher-Powell algorithm. A f t e r evaluating
t h e c o e f f i c i e n t s , t h e approximate t r a n s f e r function
o r l i f t d e f i c i e n c y f u n c t i o n becomes
.....
It c a n b e s e e n from Eq. (25) t h a t t h e r o o t s of
t h e second and t h i r d t e r m s i n b o t h numerator and
denominator a r e complex r o o t s , i n d i c a t i n g t h a t t h e
t r a n s f e r f u n c t i o n h a s complex z e r o s and p o l e s . The
complex p o l e s a r e -0.101 2 i 0.351 and -0.4465 2
i 0.601 r e s p e c t i v e l y , w h i l e t h e complex z e r o s a r e
The pre-0.0655 2 i 0.339 a n d -0.312 ? i 0.658.
s e n c e of t h e s e complex p o l e s and z e r o s i s i n d i c a t e d
by t h e p e a k s and v a l l e y s i n t h e Bode p l o t f o r
Loewy 's e x a c t l i f t d e f i c i e n c y f u n c t i o n .
The a p p r o x i m a r e t r a n s f e r f u n c t i o n , g i v e n i n Eq.
( 2 5 ) , c a n b e s u b s t i t u t e d i n Eq. (26) f o r C ' .
Applying p a r t i a l f r a c t i o n s and u s i n g a t a b l e of
inverse Laplace transforms, t h e approximate rotarywing i n d i c i a l r e s p o n s e f u n c t i o n can b e o b t a i n e d .
I n t h i s case, t h e approximate i n d i c i a l response
f u n c t i o n i s g i v e n by
.....
I n F i g . 8 , t h e r e a l and imaginary p a r t s of t h e
a p p r o x i m a t e l i f t d e f i c i e n c y f u n c t i o n , g i v e n by
Eq. ( 2 5 ) , a r e p l o t t e d t o g e t h e r w i t h t h e e x a c t
v a l u e s of F ' and G'.
I t c a n b e s e e n from F i g . 8
t h a t t h e approximate t r a n s f e r function reproduces
e x a c t l y t h e o s c i l l a t o r y n a t u r e of Loewy's l i f t
d e f i c i e n c y f u n c t i o n . From F i g . 8 i t i s e v i d e n t
t h a t b o t h t h e r e a l and imaginary p a r t s of t h e app r o x i m a t i o n t o Loewy's l i f t d e f i c i e n c y f u n c t i o n ,
The
d e v i a t e from t h e e x a c t v a l u e s f o r k > 0.70.
i m p o r t a n t i t e m t o n o t e however i s t h e a b i l i t y of
t h e approximate f u n c t i o n t o capture t h e o s c i l l a t o r y
b e h a v i o r of t h e e x a c t f u n c t i o n . The a p p r o x i m a t i o n
o b t a i n e d i n t h i s c a s e i s much b e t t e r t h a n t h e app r o x i m a t i o n o b t a i n e d i n R e f . 1 2 , w i t h t h e same
[ 5 , 5 ] o r d e r Pade a p p r o x i m a t i o n f o r t h e t r a n s f e r
f u n c t i o n . Even b e t t e r a p p r o x i m a t i o n s c o u l d b e
-0.101-r
+
e
+
e - 0 . 4 4 6 T ( ~ . 2 6 4 c o s 0 . 6 0 1 ~- 0.22 s i n 0 . 6 0 1 ~ )
(27)
(0.177 c o s 0.3511 - 0.073 s i n 0 . 3 5 1 ~ )
The i n d i c i a l r e s p o n s e f u n c t i o n $ R . W . ( ~ )i s
p l o t t e d i n F i g . 1 0 . It c a n b e s e e n from F i g . 1 0
t h a t t h e i n d i c i a l r e s p o n s e i s o s c i l l a t o r y and a t
c e r t a i n v a l u e s of T , t h e r e s p o n s e o v e r s h o o t s t h e
steady v a l u e 1.0. This r e s u l t i n d i c a t e s t h a t f o r
a s t e p change i n a n g l e of a t t a c k of t h e r o t o r
b l a d e , t h e t h r u s t developed by t h e r o t o r w i l l overshoot t h e steady v a l u e before f i n a l l y reaching t h e
s t e a d y v a l u e . T h i s phenomenon h a s been experimenta l l y observed i n Ref. 1 4 . The a u t h o r s of R e f . 14
n o t e d t h a t f o r a r a p i d change i n c o l l e c t i v e p i t c h
s e t t i n g of a model r o t o r , t h e measured t h r u s t
o v e r s h o o t s t h e s t e a d y v a l u e of t h e t h r u s t . T h i s
i s p r e c i s e l y t h e t r e n d e v i d e n t from t h e i n d i c i a l
r e s p o n s e f u n c t i o n ' R . ~ . ( T ) g i v e n by Eq. (27) and
a l s o shown i n F i g . 1 0 . F u r t h e r m o r e , i t should b e
noted t h a t t h i s is t h e f i r s t time i n t h e l i t e r a t u r e
t h a t a rotary-wing i n d i c i a l r e s p o n s e f u n c t i o n h a s
been p r e s e n t e d .
I n g e n e r a l , t h e r o t a r y wing i n d i c i a l r e s p o n s e
function can be w r i t t e n a s
-b.T
N
Z
m R e W . ( ~2) 1 . 0 -
[Aj c o s w.r
J
e
j=l
+ Bj
s i n w.r]
J
(28)
The v a l u e s of A j , Bj and b . ' s depend on t h e r o t o r
J
geometry and t h e r o t o r o p e r a t i n g c o n d i t i o n s . They
c a n b e e v a l u a t e d from t h e a p p r o x i m a t e t r a n s f e r
f u n c t i o n c o r r e s p o n d i n g t o Loewy ' s l i f t d e f i c i e n c y
function, defined f o r t h a t s p e c i f i c r o t o r .
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R e c a l l from t h e p r e v i o u s d i s c u s s i o n of t h e
f i x e d wing c a s e t h a t t h e i n d i c i a l r e s p o n s e f u n c t i o n
can be w r i t t e n a s
~I
Comparing t h e r o t a r y wing i n d i c i a l r e s p o n s e f u n c t i o n s $IR.w., Eq. ( 2 8 ) , w i t h t h e f i x e d wing i n d i c i a l
response function
Eq. ( 2 9 ) , it c a n b e conc l u d e d t h a t t h e r o t a r y wing i n d i c i a l r e s p o n s e
f u n c t i o n i s q u a l i t a t i v e l y d i f f e r e n t from t h e f i x e d
wing i n d i c i a l r e s p o n s e f u n c t i o n . The r o t a r y wing
i n d i c i a l response function e x h i b i t s an o s c i l l a t o r y
n a t u r e whereas t h e f i x e d wing i n d i c i a l r e s p o n s e
is non-oscillatory.
aF.w.,
The Second Example
The p r e s c r i b e d p a r a m e t e r s f o r t h i s c a s e a r e :
The l o c a t i o n of t h e t y p i c a l b l a d e s e c t i o n i s
s e l e c t e d t o b e a t 0.75R.
The b l a d e semichord i n
b l =~ 0.024.
F i g u r e 11 i l l u s t r a t e s t h e r e a l (F') and
imaginary ( G I ) p a r t s of Loewy's e x a c t l i f t
d e f i c-i e n c y f u n c t i o n f o r t h e v a l u e s of 5, = 3.2725
and re = 7.8125.
The r e a l and imaginary p a r t s of
t h e Loewy's l i f t d e f i c i e n c y f u n c t i o n a r e h i g h l y
o s c i l l a t o r y . The Bode p l o t of Loewy's l i f t def i c i e n c y f u n c t i o n i s p r e s e n t e d i n F i g . 12. The
Bode p l o t h a s many p e a k s and v a l l e y s . Hence t h e
corresponding approximate l i f t d e f i c i e n c y o r transf e r f u n c t i o n which we s e e k , must have a s many
complex p o l e s and complex z e r o s a s t h e r e a r e peaks
and v a l l e y s , i n t h e Bode p l o t .
I n Fig. 12, t h e
odd numbers, c o r r e s p o n d i n g t o t h e v a l l e y s i n d i c a t e
complex z e r o s and t h e even numbers c o r r e s p o n d i n g t o
t h e p e a k s i n d i c a t e complex p o l e s . For p r a c t i c a l
s i t u a t i o n s , h i g h reduced f r e q u e n c i e s above k 2 1 ,
a r e n o t common i n rotary-wing a p p l i c a t i o n s . Theref o r e t h e approximate t r a n s f e r f u n c t i o n , i n t h i s
case, is constructed so a s t o capture t h e f i r s t
e i g h t complex p o l e s and complex z e r o s . T h i s
approximate t r a n s f e r f u n c t i o n can be w r i t t e n i n
8
n
8
I( i k ) ' + a 2 j it + a 2 j + l ~
[(ik12+b
ik+b2j+1~
2j
j=1
*symbol i n d i c a t e s m u l t i p l i c a t i o n
T a b l e 11, i n Ref. 1 6 , g i v e s a l l t h e p o l e s and
z e r o s of t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n r e p r e s e n t e d by Eq. ( 3 0 ) . I t c a n b e s e e n t h a t t h e r e a r e
e i g h t complex p o l e s and e i g h t complex z e r o s , i n
a d d i t i o n t o one r e a l p o l e and a r e a l z e r o . The
d e t a i l s of t h e c a l c u l a t i o n s of t h e p o l e s and z e r o s
a r e g i v e n i n Ref. 16.
I n F i g . 1 1 , t h e r e a l and imaginary p a r t s of
t h e approximate l i f t d e f i c i e n c y o r t r a n s f e r function,
a r e shown t o g e t h e r w i t h Loewy's e x a c t l i f t def i c i e n c y f u n c t i o n . It can be seen t h a t t h e approximate t r a n s f e r f u n c t i o n f o l l o w s Loewy's e x a c t l i f t
d e f i c i e n c y f u n c t i o n and t h e agreement between t h e
two c u r v e s i s v e r y good. The i n d i c i a l r e s p o n s e
f u n c t i o n obtained using t h i s approximate t r a n s f e r
r e s p o n s e f u n c t i o n 1 6 o v e r s h o o t s t h e s t e a d y v a l u e of
t h e r e s p o n s e a t c e r t a i n v a l u e s of t i m e T and t h e
i n d i c i a l response function i s q u a l i t a t i v e l y similar
t o t h e i n d i c i a l response function obtained f o r t h e
f i r s t example shown i n F i g . 10.
F i n a l l y , it should be noted t h a t i n t h i s study
Loewy's l i f t d e f i c i e n c y f u n c t i o n i n i t s o r i g i n a l
form was u s e d . However i t h a s been shown i n R e f s .
17 and 18 t h a t t h e z e r o f r e q u e n c y l i m i t of Loewy's
l i f t deficiency f u n c t i o n does not approach u n i t y
a s t h e reduced f r e q u e n c y a p p r o a c h e s z e r o .
4.
Concluding Remarks
I n u n s t e a d y aerodynamic t h e o r i e s , t h e l i f t
d e f i c i e n c y f u n c t i o n p l a y s t h e r o l e of a n aerodynamic
t r a n s f e r f u n c t i o n r e l a t i n g t h e 314-chord downwash
v e l o c i t y t o t h e l i f t on t h e a i r f o i l . T h e r e f o r e ,
t h e Bode p l o t of t h e l i f t d e f i c i e n c y f u n c t i o n can
b e used t o o b t a i n i m p o r t a n t i n f o r m a t i o n on t h e
q u a l i t a t i v e n a t u r e of t h e p o l e s and z e r o s a s w e l l a s
t h e i r l o c a t i o n s . Using t h i s i n f o r m a t i o n , a p p r o x i mate f i n i t e s t a t e l i f t clef i c i e n c y f u n c t i o n s were
s u c c e s s f u l l y f o r m u l a t e d f o r b o t h f i x e d wing and
r o t a r y wing a p p l i c a t i o n s , such a s T h e o d o r s e n ' s
and Loewy's l i f t d e f i c i e n c y f u n c t i o n s .
Based upon t h e a p p r o x i m a t e t r a n s f e r f u n c t i o n
obtained f o r t h e l i f t deficiency functions, t h e
It
i n d i c i a l r e s p o n s e f u n c t i o n s were e v a l u a t e d .
was found t h a t t h e r o t a r y wing i n d i c i a l r e s p o n s e
f u n c t i o n h a s a n o s c i l l a t o r y n a t u r e , and which c a u s e s
t h e i n d i c i a l response t o overshoot i t s steady s t a t e
v a l u e before reaching i t . This behavior i s a l s o
c o n s i s t e n t w i t h e x p e r i m e n t a l e v i d e n c e l h . On t h e
o t h e r hand, t h e f i x e d wing i n d i c i a l r e s p o n s e i s
non-oscillatory.
F u r t h e r m o r e , i t s h o u l d b e emphasized t h a t t h i s i s t h e f i r s t time t h a t a rotarywing i n d i c i a l r e s p o n s e f u n c t i o n h a s been p r e s e n t e d
in the literature.
The f i n i t e s t a t e u n s t e a d y aerodynamic model,
o b t a i n e d i n t h e p r e s e n t s t u d y , h a s a number of
i m p o r t a n t p o t e n t i a l a p p l i c a t i o n s i n rotary-wing
a e r o e l a s t i c s t a b i l i t y and r e s p o n s e s t u d i e s , such
as:
( a ) rotary-wing a e r o e l a s t i c s t a b i l i t y and
r e s p o n s e problems i n f o r w a r d f l i g h t , (b) s i m u l a t i o n
of s u b c r i t i c a l f l u t t e r t e s t i n g of r o t o r s where t h e
d e t e r m i n a t i o n of damping l e v e l s i s i m p o r t a n t b e f o r e
a c t u a l f l u t t e r boundaries a r e encountered, (c)
t r e a t m e n t of rotary-wing a e r o e l a s t i c systems w i t h
a c t i v e f e e d b a c k c o n t r o l systems, such a s h i g h e r
harmonic c o n t r o l s .
B i s p l i n g h o f f , R.L., A s h l e y , H. and Halfman,
R.L., A e r o e l a s t i c i t y , Addison-Wesley, 1955.
Acknowledgement
The a u t h o r s would l i k e t o t h a n k M r . V .
Ramanarayanan, C a l i f o r n i a I n s t i t u t e of Technology
f o r t h e d i s c u s s i o n s r e g a r d i n g t h e m o d e l l i n g of
t r a n s f e r f u n c t i o n s based on t h e Bode p l o t .
.
Venkatesan, C . and Friedmann, P .P , " F i n i t e
S t a t e Modelling of Unsteady Aerodynamics and
I t s A p p l i c a t i o n t o a R o t o r Dynamic Problem",
U n i v e r s i t y of C a l i f o r n i a , School of E n g i n e e r i n g
and Applied S c i e n c e R e p o r t , UCLA-ENG-85-10,
March 1985.
References
Theodorsen, T., "General Theory of Aerodynamic
I n s t a b i l i t y and t h e Mechanism of F l u t t e r " ,
NACA R e p o r t 496, 1935.
Ashgar-Hessari-Dinyavar i, M., "Unsteady Aerodynamics i n t h e Time and Frequency Domains f o r
Finite-Time A r b i t r a r y Motion of H e l i c o p t e r
Rotor B l a d e s i n Hover and i n Forward F l i g h t " ,
Ph.D. D i s s e r t a t i o n , Mechanical, Aerospace and
N u c l e a r E n g i n e e r i n g Department, U n i v e r s i t y of
C a l i f o r n i a , Los Angeles, C a l i f o r n i a , March
1985.
Downloaded by UNIVERSITY OF ADELAIDE on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1986-865
Greenberg, J . M . , " A i r f o i l i n S i n u s o i d a l Motion
i n P u l s a t i n g Stream", NACA TN 1326, 1947.
Loewy, R.G., "A Two-Dimensional Approximation
of Unsteady Aerodynamics of R o t a r y Wings",
J o u r n a l of t h e A e r o n a u t i c a l S c i e n c e s , Vol. 24,
No. 2 , F e b r u a r y 1957.
D i n y a v a r i , M.A.H. and Friedmann, P.P., " F i n i t e
Time A r b i t r a r y Motion Unsteady Cascade A i r f o i l Theory f o r H e l i c o p t e r R o t o r s i n Hover",
P r o c e e d i n g s of t h e E l e v e n t h European Rotorc r a f t Forum, P a p e r No. 26, London, England,
September 1985, pp. 26.1-26.32.
Shipman, K.W. and Wood, E.R., "A Two-Dimens i o n a l Theory f o r R o t o r B l a d e F l u t t e r i n
Forward F l i g h t " , J o u r n a l of A i r c r a f t , Vol. 8 ,
No. 1 2 , December 1971.
G a r r i c k , G . E . , "On Some R e c i p r o c a l R e l a t i o n s
i n t h e Theory of N o n - s t a t i o n a r y Flows", NACA
Report 629, 1938.
J o n e s , R.T., "The Unsteady L i f t of a Wing of
F i n i t e Aspect R a t i o " , NACA R e p o r t 681, 1940.
S e a r s , W .R., " O p e r a t i o n a l Methods i n t h e
Theory of A i r f o i l s i n Non-Uniform Motion",
J o u r n a l of F r a n k l i n I n s t i t u t e , Vol. 230, No. 1 ,
J u l y 1940.
Edwards, 3 .W., "Unsteady Aerodynamic Modell i n g and A c t i v e A e r o e l a s t i c C o n t r o l " , SUDMR
504, S t a n f o r d U n i v e r s i t y , F e b r u a r y 1977.
Vepa, R., "On t h e Use of Pade Approximants t o
R e p r e s e n t Unsteady Aerodynamic Loads f o r
A r b i t r a r y Small Motions of Wings", AIAA p a p e r
76-17, AIAA 1 4 t h Aerospace S c i e n c e s Meeting,
Washington, D . C . , J a n u a r y 26-28, 1976.
REFERENCE AIRFOIL
0
Dowell, E.H., "A Simple Method f o r C o n v e r t i n g
Frequency Domain Aerodynamics t o t h e Time
Domain", NASA TM 81844, 1980.
J o h n s o n , W . , H e l i c o p t e r Theory, P r i n c e t o n
U n i v e r s i t y P r e s s , 1980.
D i n y a v a r i , M.A.H. and Friedmann, P . P . , "Uns t e a d y Aerodynamics i n Time and Frequency
Domains f o r F i n i t e Time A r b i t r a r y Motion of
R o t a r y Wings i n Hover and Forward F l i g h t " ,
AIAA P a p e r 84-0988, P r o c e e d i n g s of t h e A I M /
ASME/ASCE/AHS,2 5 t h S t r u c t u r e s , S t r u c t u r a l
Dynamics and M a t e r i a l s Conference, May 14-16,
Palm S p r i n g s , C a l i f o r n i a , pp. 266-282, 1984.
Ogata, K . , Modern C o n t r o l E n g i n e e r i n g ,
P r e n t i c e - H a l l , 1970.
C a r p e n t e r , P . J . and F r i d o v i t c h , B . , " E f f e c t
of Rapid B l a d e - P i t c h I n c r e a s e on t h e T h r u s t
and I n d u c e d - V e l o c i t y Response of a F u l l
S c a l e H e l i c o p t e r v ' , NACA TN 3044, November
0
1
0
2
1
0
?
1
=,:,
Fig. 1
q=&l
- -q = BLADE NUMBER
n = ROTOR REVOLUTION INDEX
Loewy's I n c o m p r e s s i b l e Aerodynamic Model
0-'24-0
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0 dbidecade
Exact Curve
al = 1 0 bl
A
O
A-A-
Asymptotes of -20 l o g l O d
Y
0 dbidecade
A
1 +
b?
Fig. 2
Bode P l o t of G(iw) =
wlth a, = 1 0 bl
rlope -40 dbidecade
l+i2Cl (w/al)+(iw/al) 2
Fig. 3
;
Bode P l o t o f G(iw) =
l+i2~2(w/bl)+(iw/bl)
a l = 10 b,
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Fig. 4
Theodorsen's L i f t D e f i c i e n c y F u n c t i o n
- - Present Analysis, Eq. (17)
0-0 R.T. J o n e s , Ref. 6
\
High Frequency Asymptote
20 loglO 0.5 = -6.02 db
--
Fig. 5
Exact Curve
Asymptotic Behavtour of Approximate Function
Bode P l o t of Theodorsen's L i f t D e f i c i e n c y
F u n c t i o n , 2 P o l e Approximation, E q . ( 1 7 )
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Fig. 6
Theodorsen's L i f t Deficiency Function
3 P o l e Approximation, Eq. (18)
Low Frequency Asymptote 20 loglOl = 0 db
0 dbldecade
0 dbldecade
-20 dbldecade
-20 dbldecade
4
-
--
Fig. 7
-
0 dbldecade
Exact Curve
Asymptotic Behaviour of Approxmate Function
Bode P l o t o f T h e o d o r s e n ' s L i f t D e f i c i e n c y F u n c t i o n ,
3 P o l e Approximation, Eq. (18)
Exact Loewy's Function
Approximate Function (Eq.28)
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Result Ref. 15
Fig. 8
Low Frequency Asymptote.
Loewy's L i f t D e f i c i e n c y F u n c t i o n and
I t s Approximation f o r Example 1
20 log,"l
=
0 db
\
0 dbldecade
Htgh Frequency Asymptote.
20 loglO 0.5 = -6.02 db
Fig. 9
Bode P l o t of Loewy's L i f t D e f i c i e n c y F u n c t i o n f o r Example 1
Exact Loewy's Function
---
Approximate Function lEq.34)
-
R o t a r y Wing I n d i c i a 1 R e s p o n s e F u n c t i o n
f o r Example 1
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Fig. 10
Fig. 11
F i g . 12
Loewy's L i f t D e f i c i e n c y F u n c t i o n a n d
I t s A p p r o x i m a t i o n f o r Example 2
Bode P l o t o f Loewy's L i f t D e f i c i e n c y F u n c t i o n f o r Example 2
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