NASA Technical Memorandum 84281
1
\
i
1
I
!
A Mathematical Model of a Single
Main Rotor Helicopter for Piloted
Simulation
Peter D. Talbot, Bruce E. Tinling,
William A. Decker, and Robert T. N. Chen
t
September 1982
NASA
National Aeronautics and
Space Administration
...
NASA Technical Memorandum 84283
A Mathematical Model of a Single
Main Rotor Helicopter for Piloted
Simulation
Peter D. Talbot
Bruce E. Tinling
William A. Decker
Robert T. N. Chen, Ames Research Center, Moffett,Field, California
National Aeronautics and
Space Administration
Ames Research Center
Moffett Field California 94035
CONTENTS
Page
...................................
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SIMULATION MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Main r o t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
T a i l rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Empennage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotor R o t a t i o n a l Degree of Freedom and RPM Governing . . . . . . . . . .
C o n t r o l Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . .
L i n e a r i z e d Six-Degree-of-Freedom Model . . . . . . . . . . . . . . . . .
SuMMAEtY
I
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0
0
r
I
c,
APPENDICES
A Notation
B
C
D
E
F
G
H
I
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Axissystems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Main Rotor F l a p p i n g Dynamics and F o r c e and Moment C a l c u l a t i o n . . . .
T a i l Rotor Flapping and Force C a l c u l a t i o n . . . . . . . . . . . . . .
Empennage F o r c e s and Moments . . . . . . . . . . . . . . . . . . . .
C a l c u l a t i o n of F u s e l a g e F o r c e s and Moments . . . . . . . . . . . . .
RPM Governor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cockpit C o n t r o l s and C y c l i c C o n t r o l Phasing . . . . . . . . . . . . .
L i n e a r i z e d Six-Degree-of-Freedom
Dynamics
1
9
13
15
23
28
32
35
37
R e p r e s e n t a t i o n of H e l i c o p t e r
.............................
J C o n f i g u r a t i o n D e s c r i p t i o n Requirements . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
k
1
39
41
46
TABLES
Page
I
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.................
1-1
Elements of the Linearized Equations of Motion
40
J-1
Configuration Description Requirements
42
V
FIGURES
Page
1
Block Diagram Showing P r i n c i p a l Elements of S i n g l e Rotor H e l i c o p t e r
M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
T y p i c a l V a r i a t i o n of Empennage L i f t and Drag C o e f f i c i e n t s
5
3
Block Diagram of RPM Governor
.......
.....................
6
4
S t r u c t u r e of C o n t r o l System Model
...................
7
B- 1
The Hub-Wind Axis System w i t h Main Rotor F o r c e , Moment, and V e l o c i t y
Components Defined
B-2
Hub-Body, A i r c r a f t
D- 1
T a i l Rotor Forces and Moments
.........................
Reference and Body-c.g. Axis Systems . . . . . . . .
.....................
vii
3
14
24
A MATHEMATICAL MODEL OF A SINGLE MAIN ROTOR HELICOPTER
FOR PILOTED SIMULATION
P e t e r D. T a l b o t , Bruce E. T i n l i n g , W i l l i a m A. Decker, and Robert T. N . Chen
Ames Research C e n t e r
SUMMARY
4
T h i s r e p o r t documents a h e l i c o p t e r mathematical model s u i t a b l e f o r p i l o t e d
s i m u l a t i o n of f l y i n g q u a l i t i e s . The mathematical model i s a n o n l i n e a r , t o t a l f o r c e
model h a s t e n d e g r e e s of
and moment model of a s i n g l e main r o t o r h e l i c o p t e r .
freedom: s i x rigid-body, t h r e e r o t o r - f l a p p i n g , and t h e r o t o r r o t a t i o n a l d e g r e e s of
freedom. The r o t o r model assumes r i g i d b l a d e s w i t h r o t o r f o r c e s and moments r a d i a l l y
i n t e g r a t e d and summed a b o u t t h e azimuth. The f u s e l a g e aerodynamic model u s e s a
d e t a i l e d r e p r e s e n t a t i o n over a nominal a n g l e of a t t a c k and s i d e s l i p range of ?15",
and i t u s e s a s i m p l i f i e d c u r v e f i t a t l a r g e a n g l e s of a t t a c k o r s i d e s l i p .
Stabilizi n g s u r f a c e aerodynamics are modeled with a l i f t c u r v e s l o p e between s t a l l l i m i t s and
a g e n e r a l c u r v e f i t f o r l a r g e a n g l e s of a t t a c k . A g e n e r a l i z e d s t a b i l i t y and c o n t r o l
augmentation system i s d e s c r i b e d . A d d i t i o n a l computer s u b r o u t i n e s p r o v i d e o p t i o n s
f o r a s i m p l i f i e d engine/governor model, atmospheric t u r b u l e n c e , and a l i n e a r i z e d
six-degree-of-freedom dynamic model f o r s t a b i l i t y and c o n t r o l a n a l y s i s .
me
I N TRODUC T I ON
An expanded f l y i n g - q u a l i t i e s d a t a b a s e i s needed f o r u s e i n d e v e l o p i n g d e s i g n
c r i t e r i a f o r f u t u r e h e l i c o p t e r s . A safe and c o s t - e f f e c t i v e way t o e s t a b l i s h s u c h a
d a t a b a s e i s t o conduct e x p l o r a t o r y i n v e s t i g a t i o n s u s i n g p i l o t e d ground-based simul a t o r s , and t h e n t o s u b s t a n t i a t e t h e r e s u l t s i n f l i g h t u s i n g v a r i a b l e s t a b i l i t y
research h e l i c o p t e r s .
A m a t h e m a t i c a l model s u i t a b l e f o r real-time p i l o t e d s i m u l a t i o n of s i n g l e r o t o r
h e l i c o p t e r s h a s been implemented a t Ames Research Center. A s d e s c r i b e d i n r e f e r ence 1, s i m u l a t i o n models used a t Ames Research Center c o n s i s t of a common c o r e of
rigid-body e q u a t i o n s and an aerodynamic model t h a t p r o v i d e s t h e aerodynamic f o r c e s
and moments. T h i s r e p o r t documents t h e e q u a t i o n s used i n t h e aerodynamic model.
*F
The r e p o r t c o n s i s t s of a b r i e f d e s c r i p t i o n of t h e o v e r a l l model and i t s compon e n t s and a p p e n d i c e s t h a t d e t a i l t h e e q u a t i o n s used i n the model and t h e p a r a m e t e r s
required t o describe a helicopter configuration.
..%$.
SIMULATION MODEL
The o v e r a l l arrangement of t h e s i m u l a t i o n model i s shown i n f i g u r e 1. The
p r i n c i p a l assumptions and c o n s i d e r a t i o n s employed i n developing each element of t h e
model a r e given i n t h e main body of t h e r e p o r t ; d e t a i l e d e q u a t i o n s f o r t h e f o r c e s
and moments a r e g i v e n i n t h e appendices. The model e l e m e n t s , denoted T i i n f i g u r e 1, are r e q u i r e d t o a c h i e v e t r a n s f e r of v e l o c i t i e s , f o r c e s , and moments from
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one a x i s system t o a n o t h e r ; i n some i n s t a n c e s , t o account f o r aerodynamic i n t e r f e r ence e f f e c t s between model components. These elements are d e s c r i b e d i n t h e append i c e s where r e q u i r e d .
i
I
The n o t a t i o n employed i s l i s t e d i n appendix A, and a d e s c r i p t i o n of t h e v a r i o u s
a x i s systems i s given i n appendix B. The v a r i a b l e names i n t h e FORTRAN coding f o r
t h e computer program have been chosen t o be e a s i l y i d e n t i f i a b l e from t h e n o t a t i o n
used h e r e i n . Through t h i s mnemonic d z v i c e , e q u a t i o n s I n the appendix can be i d e n t i f i e d i n the computer program l i s t i n g s .
I
Main Rotor
?A
r,
The development of t h e e q u a t i o n s d e s c r i b i n g t h e dynamics and t h e f o r c e s and
moments a c t i n g on t h e main r o t o r are given i n d e t a i l i n r e f e r e n c e s 2 and 3 . T h i s
mathematical r e p r e s e n t a t i o n e x p l i c i t l y a c c o u n t s f o r t h e dynamic e f f e c t of r o t o r modes,
such as r o t o r - b l a d e f l a p p i n g , which can b e i n a frequency r a n g e which i s i m p o r t a n t i n
s t u d i e s of f l y i n g q u a l i t i e s . For t h e r o t o r model d e s c r i b e d i n t h i s r e p o r t , t h e f l a p p i n g dynamics were approximated u s i n g a t i p - p a t h p l a n e r e p r e s e n t a t i o n .
The f l a p p i n g e q u a t i o n of motion of t h e r o t o r b l a d e w a s f i r s t developed u s i n g t h e
f o l l o w i n g assumptions. The assumptions are s i m i l a r t o t h o s e used f o r t h e "classicall'
e q u a t i o n s ( r e f s . 4 and 5).
1. The r o t o r b l a d e w a s r i g i d i n bending and t o r s i o n , and t h e t w i s t of t h e b l a d e
was linear.
2. The f l a p p i n g a n g l e and i n f l o w a n g l e were assumed t o b e s m a l l and t h e analys i s u t i l i z e d a simple s t r i p t h e o r y .
3 . The e f f e c t s of t h e a i r c r a f t motion on t h e b l a d e f l a p p i n g were l i m i t e d t o
t h e a n g u l a r r a t e p and q , and t h e
t h o s e due t o t h e a n g u l a r a c c e l e r a t i o n
and
normal a c c e l e r a t i o n .
4,
4 . The r e v e r s e d flow r e g i o n w a s ignored and t h e c o m p r e s s i b i l i t y and s t a l l
e f f e c t s disregarded.
>
5.
The i n f l o w w a s assumed t o b e uniform and no i n f l o w dynamics were used.
6.
The t i p - l o s s f a c t o r was assumed t o b e 1.
The f l a p p i n g e q u a t i o n s of motion e x p l i c i t l y c o n t a i n t h e primary d e s i g n parame t e r s , namely: f l a p p i n g h i n g e r e s t r a i n t , h i n g e o f f s e t , b l a d e Lock number, and p i t c h f l a p c o u p l i n g . The b l a d e f l a p p i n g i n t h o s e e q u a t i o n s w a s then approximated by t h e
f i r s t harmonic terms w i t h time-varying c o e f f i c i e n t s , t h a t i s ,
7.
B(t) = a o ( t ) - a l ( t ) c o s $
-
b l ( t ) s i n $.
I n t h e developmenk of t h e e q u a t i o n s f o r f o r c e s and moments, t h e same s e t of
b a s i c a s s u m p t i m s ( 1 through 7 above) , d i s c u s s e d i n c o n j u n c t i o n w i t h t h e development
of t h e t i p - p a t h p l a n e dynamic e q u a t i o n s , w a s u t i l i z e d . 'rhus, aerc,dynamically,
momentum t h e o r y w a s used i n c o n j u n c t i o n w i t h t h e uniform i n f l o w ; s i m p l e s t r i p t h e o r y
w a s u t i i i z e d and t h e b l a d e f o r c e s were a n a l y t i c a l l y i n t e g r a t e . 1 over t h e r a d i u s .
Because t h e r e v e r s e d flow r e g i o n and the s t a l l and c o m p r e s s i b i l i t y e f f e c t s w e r e
i g n o r e d , t h e t o t a l r o t o r f o r c e s and moments were a g a i n a n a l y t i c a L l y o b t a i n e d by
3
summing t h e c o n t r i b u t i o n s , t o each b l a d e , that w e r e a n a l y t i c a l f u n c t i o n s of t h e
azimuth. Because of t h e s e assumptions and s i m p l i f i c a t i o n s , t h e r e s u l t s of the analysis are v a l i d only f o r a l i m i t e d r a n g e of f l i g h t c o n d i t i o n s . N e v e r t h e l e s s , a p r e v i o u s s t u d y ( r e f . 5) has shown t h a t t h i s t y p e of a n a l y s i s i s v a l i d f o r s t a b i l i t y and
c o n t r o l i n v e s t i g a t i o n s of t h e r o t o r c r a f t up t o an advance r a t i o of about 0.3.
Also,
s i m i l a r t o t h e development of t h e t i p - p a t h p l a n e dynamic e q u a t i o n s , t h e s e r o t o r
f o r c e s and moments were f i r s t o b t a i n e d i n t h e wind-hub c o o r d i n a t e system. They were
then transformed i n t o t h e hub-body system ( s e e appendix C ) .
The f o r c e s and moments t h u s developed c o n t a i n p e r i o d i c terms; t h e h i g h e s t harmonic terms correspond d i r e c t l y t o t h e number of r o t o r b l a d e s . For example, f o r a
three-bladed r o t o r , t h e f o r c e and moment e q u a t i o n s c o n t a i n o n l y t h r e e / r e v o l u t i o n
harmonic terms, and f o r a four-bladed r o t o r , f o u r / r e v o l u t i o n harmonic terms. The
frequency of t h e s e harmonic terms i s s u f f i c i e n t l y h i g h t o b e of n o i n t e r e s t t o hand l i n g q u a l i t y i n v e s t i g a t i o n s . These t e r m s have t h e r e f o r e been d e l e t e d . The r e s u l t i n g f o r c e and moment e x p r e s s i o n s a r e g i v e n i n appendix C.
3
-'
A development t o modify t h e s e f o r c e s and moment e q u a t i o n s t o i n c l u d e t h e e f f e c t s
of nonuniform i n f l o w , similar t o t h e development f o r t h e f l a p p i n g e q u a t i o n s i n r e f e r e n c e s 6 and 7 , i s i n p r o g r e s s . These e q u a t i o n s , a l o n g w i t h t h e modified t i p - p a t h
p l a n e r e p r e s e n t a t i o n g i v e n i n r e f e r e n c e 7 w i l l l a t e r s u p e r s e d e t h o s e shown i n
appendix C.
T a i l Rotor
The t a i l r o t o r w a s modeled a s a t e e t e r i n g r o t o r w i t h o u t c y c l i c p i t c h . For t h i s
c a s e , t h e f o r c e s i n t h e wind-hub system may b e o b t a i n e d from t h e e x p r e s s i o n s d e r i v e d
f o r t h e main r o t o r by s e t t i n g t h e l a t e r a l and l o n g i t u d i n a l c y c l i c p i t c h terms (Alc
and Blc)
equal t o zero.
F u r t h e r , s i n c e t h e t a i l r o t o r f l a p p i n g frequency i s much
h i g h e r than t h a t of t h e main r o t o r system, t h e t i p - p a t h p l a n e dynamics may b e
n e g l e c t e d . Thus, f o r t a i l r o t o r a p p l i c a t i o n s , t h e f i r s t and second d e r i v a t i v e s of
t h e b l a d e - f l a p p i n g n o n r o t a t i n g c o o r d i n a t ..
e s are set e q u a l t o z e r o i n t h e f o r c e equat i o n s (io = i 1 = 61 = 0 and 20 = 81 = b l = 0 ) . The r e s u l t i s a s e t of b a s i c q u a s i s t a t i c f o r c e e x p r e s s i o n s s i m i l a r t o t h o s e i n c l a s s i c a l work ( r e f s . 3 and 4 ) .
The l o c a l flow a t t h e t a i l r o t o r i n c l u d e s t h e e f f e c t of downwash from t h e main
r o t o r system. The method employed t o e s t i m a t e t h i s downwash and t h e e q u a t i o n s f o r
t a i l r o t o r f o r c e s a r e g i v e n i n appendix D.
Empennage
-3
The l i f t and d r a g f o r c e s on t h e v e r t i c a l f i n and h o r i z o n t a l t a i l are approximated f o r a l l a n g l e of a t t a c k and s i d e s l i p , i n c l u d i n g rearward f l i g h t . P r o v i s i o n i s
made f o r t h e a d d i t i o n of terms due t o main r o t o r - i n d u c e d v e l o c i t i e s a t t h e h o r i z o n t a l
t a i l , and t a i l r o t o r v e l o c i t i e s a t t h e v e r t i c a l f i n .
The p r i n c i p a l assumptions made i n d e v e l o p i n g t h e e x p r e s s i o n s f o r t h e f o r c e s and
moments due t o t h e v e r t i c a l f i n and h o r i z o n t a l t a i l are as f o l l o w s :
1. The l i f t and d r a g f o r c e s a r e a p p l i e d a t t h e q u a r t e r chord of each s u r f a c e
a t t h e spanwise l o c a t i o n of t h e c e n t e r of area.
4
-:
I
-I
2.
-
The a i r f o i l p r o f i l e s are symmetrical.
3. The l i f t c u r v e s l o p e p r i o r t o s t a l l i s g i v e n by s i m p l e l i f t i n g - l i n e t h e o r y
assuming an e l l i p t i c a l l i f t d i s t r i b u t i o n w i t h uniform downwash. C o r r e c t i o n s are
a p p l i e d f o r s i d e s l i p and f o r sweep of t h e v e r t i c a l f i n .
4 . Maximum l i f t c o e f f i c i e n t i s s p e c i f i e d ; however, i f t h e l i f t c u r v e i ss l oassumed
pe is
such t h a t t h i s v a l u e i s n o t reached a t a n a n g l e of attack of ~ / 4 C, b a x
i
t o o c c u r a t t h i s a n g l e of attack.
~
I
c
.
5.
P o s t s t a l l v a r i a t i o n of l i f t c o e f f i c i e n t i s based on
Cbax
d e c r e a s i n g by
20% as t h e a n g l e of a t t a c k i s i n c r e a s e d by 20%, and f o l l o w i n g a p a r t i c u l a r v a r i a t i o n
t h e r e a f t e r t o r e a c h z e r o l i f t a t an a n g l e of a t t a c k of ~ / 2 .
6.
L i f t c o e f f i c i e n t i n rearward f l i g h t i s 80% of t h a t i n forward f l i g h t .
The p r o f i l e drag c o e f f i c i e n t v a r i e s w i t h a n g l e of a t t a c k and r e a c h e s a
7.
v a l u e of 1 when a = 2 7 ~ 1 2 .
8. The induced d r a g c o e f f i c i e n t v a r i e s a s t h e s q u a r e of t h e c a l c u l a t e d l i f t
coefficient .
A t y p i c a l v a r i a t i o n of t h e l i f t and d r a g c o e f f i c i e n t s f o r an empennage s u r f a c e
i s shown i n f i g u r e 2. Expressions f o r c a l c u l a t i n g t h e empennage f o r c e s and t h e
r e q u i r e d t r a n s f o r m a t i o n s f o r v e l o c i t i e s and f o r c e s a r e g i v e n i n appendix E.
F i g u r e 2 . - T y p i c a l v a r i a t i o n of empennage l i f t and d r a g c o e f f i c i e n t s .
Fuselage
The aerodynamic model of t h e f u s e l a g e must f u l f i l l two requirements. The f i r s t
i s t o p r o v i d e an e s t i m a t e of t h e f o r c e s and moments a t s m a l l a n g l e s of a t t a c k and
5
s i d e s l i p t h a t w i l l b e encountered a t s u b s t a n t i a l forward speeds. T h i s p r o v i d e s a
r e p r e s e n t a t i o n of t h e important e f f e c t s of f u s e l a g e aerodynamics on performance and
s t a b i l i t y a t t h e s e speeds. The second requirement i s t o p r o v i d e a c o n t i n u o u s variat i o n of f o r c e s and moments over t h e e n t i r e r a n g e of a n g l e of a t t a c k and s i d e s l i p
(0" t o +180°) t h a t can be encountered i n approach t o h o v e r i n g f l i g h t o r i n hover.
C o n t i n u i t y i s r e q u i r e d t o avoid sudden u n r e a l i s t i c l i n e a r o r a n g u l a r a c c e l e r a t i o n s
i n r e s p o n s e t o a small change i n a t t i t u d e . Accuracy of t h e model a t extreme a t t i t u d e s i s c o n s i d e r e d t o be of secondary importance, s i n c e t h e f u s e l a g e f o r c e s a t t h e s e
speeds are v e r y s m a l l compared t o t h e r o t o r f o r c e s .
A t e c h n i q u e has been developed t o p r o v i d e a c o n t i n u o u s model by f i t t i n g t y p i tal v a r i a t i o n s of t h e f o r c e s and moments through d a t a p o i n t s o b t a i n e d a t s p e c i f i c
widely s e p a r a t e d a n g l e s of a t t a c k and s i d e s l i p i n a wind t u n n e l ( s e e r e f . 8 ) . Howe v e r , even t h i s s p a r s e l e v e l of t e s t d a t a f o r t h e f u s e l a g e i s t y p i c a l l y u n a v a i l a b l e
and a n a l t e r n a t i v e t e c h n i q u e must b e employed.
The model employed h e r e i n r e l i e s on s e p a r a t e r e p r e s e n t a t i o n s f o r a n g l e s of
a t t a c k and s i d e s l i p i n t h e range from -15" t o 15' and from 230" t o +180". C o n t i n u i t y
i s provided by a l i n e a r i n t e r p o l a t i o n f o r f o r c e s and moments i n t h e a n g l e range n o t
covered. The f o r c e s and moments f o r t h e lower a n g l e r a n g e are o b t a i n e d from t e s t
d a t a o r from estimates based on d a t a from similar f u s e l a g e s .
The d a t a f o r t h e h i g h
a n g l e of a t t a c k and s i d e s l i p range are based on t h e e s t i m a t e d magnitude and l o c a t i o n
of t h e d r a g f o r c e v e c t o r when t h e f u s e l a g e i s i n a 90" c r o s s f l o w , and on an approximation t o i t s observed v a r i a t i o n s w i t h a t t i t u d e from wind-tunnel tests of b o d i e s of
revolution.
Details of t h e procedure f o r e s t i m a t i n g f u s e l a g e f o r c e s and moments are g i v e n i n
appendix F.
Rotor R o t a t i o n a l Degree of Freedom and RPM Governing
An o p t i o n i s a v a i l a b l e which p r o v i d e s f o r a r o t a t i o n a l d e g r e e of freedom f o r t h e
r o t o r . When t h i s o p t i o n i s u s e d , t h e main r o t o r and t h e t a i l r o t o r r o t a t i o n a l s p e e d s
vary a c c o r d i n g t o t h e c u r r e n t t o r q u e r e q u i r e m e n t s and t h e e n g i n e power a v a i l a b l e .
The i n i t i a l trim c o n d i t i o n s e s t a b l i s h b a s e l i n e v a l u e s of r o t o r speeds and e n g i n e
t o r q u e . D e v i a t i o n s from t h e s e b a s e l i n e v a l u e s change the r o t o r t o r q u e r e q u i r e m e n t s
and, hence, r o t o r s p e e d . These changes i n speed c a u s e t h e rpm governor t o v a r y t h e
f u e l f l o w t o t h e engine t o change t h e power t o m a i n t a i n t h e d e s i r e d a n g u l a r rate. A
b l o c k diagram of t h e r p m governor i s shown i n f i g u r e 3 . F u r t h e r d e t a i l s of the
dynamic model f o r t h i s d e g r e e of freedom and t h e rpm governor a r e given i n
appendix G.
/
(MAIN AND TAIL ROTOR
Q ~ o
THROTTLE
TORQUE REQUIRED)
(0N/O FF)
4
A
-Ai2
GOVERNOR
G (SI
%ET
=a,
*
W
KE
V
AHP
-m
550 wf
s2
-An
As2
4
s1
+
F i g u r e 3 . - Block diagram of rpm governor.
6
b
?
C o n t r o l Systems
The h e l i c o p t e r model h a s a g e n e r a l i z e d c o n t r o l system t h a t a c c e p t s i n p u t s from
t h e p i l o t , f a c i l i t a t e s c o n t r o l augmentation and s t a b i l i t y augmentation, and p r o v i d e s
f o r mechanical c o n t r o l mixing o r phasing of t h e c y c l i c i n p u t s . A b l o c k diagram of
t h i s system i s shown i n f i g u r e 4 . The c o n t r o l augmentation system employs a
*
6
4
~
f
6
6ap
-
C p j K8
XO
13
eP4 K M 3 b
+ DIRECTIONAL
,
LATERAL CYCLIC CONTROL
LONGITUDINAL CYCLIC CONTROL
COLLECTIVE CONTROL
CYCLIC
CONTROL +
PHASING
-,AIRCRAFT
AND
*DYNAMICS
GEARING
AND
+
RIGGING
AIRCRAFT
STATE, X
XO
CROSS-FEED AND
FEED FORWARD GAINS
-
FEEDBACK GAINS
XO
X = (u, w, 0 , 8 , v, p, 4, r ) T
F i g u r e 4 . - S t r u c t u r e of c o n t r o l system model.
c-
s t r u c t u r e t h a t p e r m i t s implementation of c r o s s f e e d s from each of t h e f o u r c o c k p i t
c o n t r o l s ; namely, l o n g i t u d i n a l and l a t e r a l c y c l i c s t i c k , c o l l e c t i v e s t i c k , and
d i r e c t i o n a l p e d a l s . The feedback g a i n s t r u c t u r e of t h e s t a b i l i t y augmentation system
p e r m i t s feedback p r o p o r t i o n a l t o any element of t h e s t a t e v e c t o r t o each of t h e f o u r
c o n t r o l s . The e n t i r e c o n t r o l s t r u c t u r e a l s o f a c i l i t a t e s g a i n s c h e d u l i n g a s f u n c t i o n s
of t h e f l i g h t p a r a m e t e r s , such a s a i r s p e e d .
3.
D e t a i l s c o n c e r n i n g t h e z e r o p o s i t i o n and s i g n c o n v e n t i o n of t h e c o c k p i t c o n t r o l s
and t h e mechanical c y c l i c c o n t r o l phasing l o g i c are g i v e n i n appendix H.
Atmospheric Turbulence
The r e p r e s e n t a t i o n of atmospheric t u r b u l e n c e i s based on t h e Dryden model and i s
d e s c r i b e d i n MIL-F-8785B ( r e f . 9 ) . The i n p u t s r e q u i r e d f o r t h i s model a r e t h e a i r c r a f t v e l o c i t y r e l a t i v e t o t h e a i r mass, t h e t u r b u l e n c e scale l e n g t h s , and t h e r m s
7
g u s t l e v e l s . For t h i s r e p r e s e n t a t i o n , scale l e n g t h and t h e wind v e l o c i t y can b e
s p e c i f i e d as f u n c t i o n s of a l t i t u d e . The rms g u s t l e v e l s are dependent upon wind
velocity.
L i n e a r i z e d Six-Degree-of-Freedom
Model
A computer s u b r o u t i n e i s a v a i l a b l e t h a t g e n e r a t e s t h e c o e f f i c i e n t s of a l i n e a r ,
f i r s t - o r d e r set of d i f f e r e n t i a l e q u a t i o n s t h a t r e p r e s e n t s t h e r i g i d body dynamics of
t h e h e l i c o p t e r f o r small p e r t u r b a t i o n s from a f i x e d o p e r a t i n g p o i n t . The p r i n c i p a l
assumption n e c e s s a r y t o g e n e r a t e t h i s l i n e a r s e t of e q u a t i o n s i s that t h e h e l i c o p t e r
i n i t i a l a n g u l a r rates a r e z e r o . The d i f f e r e n t i a l e q u a t i o n s are of t h e form:
k
=
[FIX
+
[GI6
where x r e p r e s e n t s p e r t u r b a t i o n s from t r i m of the s t a t e v a r i a b l e s u , w, q , 8 , v ,
p , $, and r ; and 6 r e p r e s e n t s t h e c o c t r o l d i s p l a c e m e n t s from t r i m A s e , A6,, Asa,
The g e n e r a t i o n of t h e F and G matrices and a d e s c r i p t i o n of each element
and A $ .
i s given i n appendix I.
8
G
APPENDIX A
NOTATION
a
b l a d e l i f t - c u r v e s l o p e , p e r rad
b l a d e c o n i n g a n g l e measured from hub p l a n e i n t h e hub-wind a x e s system, r a d
l o n g i t u d i n a l f i r s t - h a r m o n i c f l a p p i n g c o e f f i c i e n t measured from t h e hub
p l a n e i n t h e wind-hub a x e s system, r a d
l o n g i t u d i n a l f i r s t - h a r m o n i c f l a p p i n g c o e f f i c i e n t measured from t h e hub
p l a n e i n t h e hub-body a x e s system, r a d
l a t e r a l c y c l i c p i t c h measured from hub p l a n e i n t h e wind-wub a x e s system,
rad
l a t e r a l c y c l i c p i t c h measured from hub p l a n e i n t h e hub-body a x e s system,
rad
AR
a s p e c t r a t i o , span2/area
bl
l a t e r a l f i r s t - h a r m o n i c f l a p p i n g c o e f f i c i e n t measured from hub p l a n e i n t h e
wind-hub a x e s system, r a d
bh
l a t e r a l f i r s t - h a r m o n i c f l a p p i n g c o e f f i c i e n t measured from hub p l a n e i n t h e
hub-body a x e s system, r a d
BL
l a t e r a l d i s t a n c e ( b u t t l i n e ) i n t h e f u s e l a g e a x e s system, m ( f t )
l o n g i t u d i n a l c y c l i c p i t c h measured from hub p l a n e i n t h e wind-hub a x e s
system, r a d
BIS
l o n g i t u d i n a l c y c l i c p i t c h measured from hub p l a n e i n t h e hub-body a x e s
system, r a d
C
blade chord, m ( f t )
cn
control gearing constants, n = 1 t o 8
CB,
CAIs)
c y c l i c control rigging constants, rad
t
drag coefficient
CL
l i f t coefficient
maximum l i f t c o e f f i c i e n t
CT
rotor thrust coefficient
D
drag force, N (lb)
, T / p (,rrR2)
9
(RR)
damping m a t r i x i n f l a p p i n g d i f f e r e n t i a l e q u a t i o n s
flapping hinge o f f s e t , m(ft)
g r a v i t a t i o n a l a c c e l e r a t i o n , m/sec2 ( f t / s e c 2 )
r o t o r f o r c e normal t o s h a f t , p o s i t i v e downwind, N ( l b )
i n c i d e n c e of v e r t i c a l f i n , p o s i t i v e f o r l e a d i n g edge t o t h e l e f t , r a d
i n c i d e n c e of h o r i z o n t a l s t a b i l i z e r , p o s i t i v e f o r l e a d i n g edge up, r a d
forward t i l t of r o t o r s h a f t w . r . t .
f u s e l a g e , p o s i t i v e forward, r a d
r o t o r b l a d e moment of i n e r t i a a b o u t f l a p p i n g h i n g e , kg-m2 ( s l u g - f t 2 )
r o t o r moment of i n e r t i a about s h a f t
f a c t o r t o account f o r f r a c t i o n of v e r t i c a l t a i l i n t a i l r o t o r wake
f l a p p i n g s p r i n g c o n s t a n t , N-m/rad
pitch-flap coupling r a t i o ,
4 tan
(lb-ft/rad)
6,
s p r i n g matrix i n f l a p p i n g d i f f e r e n t i a l equations
f u s e l a g e r o l l i n g moment, n-m (f t - l b )
f u s e l a g e l i f t , N (lb)
r o l l i n g moment, p i t c h i n g moment, and yawing moment, r e s p e c t i v e l y , N-m
(ft-lb)
r o t o r b l a d e mass, kg ( s l u g s )
b l a d e weight moment about f l a p p i n g h i n g e , N-m
(lb-ft)
number of b l a d e s
r o l l , p i t c h , and yaw r a t e s i n t h e body-c.g.
a x e s system, r a d / s e c
r o l l , p i t c h , and yaw r a t e s i n t h e body-c.g.
mass, r a d / s e c
a x e s system r e l a t i v e t o t h e a i r
r o l l , p i t c h , and yaw r a t e s i n t h e hub-body a x e s system, r a d / s e c
r a t i o of f l a p p i n g frequency t o r o t o r system a n g u l a r v e l o c i t y
dynamic p r e s s u r e ,
1
2 pV2,
N/m2
(lb/ft2)
10
.
a
Q
t o r q u e , N-m ( f t - l b )
R
rotor radius, m ( f t )
S
Laplace v a r i a b l e
S
area of s t a b i l i z i n g s u r f a c e , m2 ( f t 2 )
STA
l o n g i t u d i n a l l o c a t i o n i n t h e f u s e l a g e a x e s system, m ( f t )
T
thrust, N (lb)
W
l o n g i t u d i n a l , l a t e r a l , and v e r t i c a l v e l o c i t i e s i n t h e body-c.g.
a x e s , m/sec ( f t / s e c )
s
%
system of
l o n g i t u d i n a l , l a t e r a l , and v e r t i c a l v e l o c i t i e s r e l a t i v e t o t h e a i r mass i n
t h e body-c.g. system of a x e s , m / s e c ( f t / s e c )
W
H
vi
main r o t o r induced v e l o c i t y a t r o t o r d i s k , m/sec ( f t / s e c )
Wf
f u e l flow rate, N/hr ( l b / h r )
WL
v e r t i c a l l o c a t i o n i n t h e f u s e l a g e a x e s system, m ( f t )
Z
l o n g i t u d i n a l , l a t e r a l , and v e r t i c a l f o r c e s i n t h e body-c.g.
N (Ib)
a
s t a b i l i z i n g s u r f a c e a n g l e of a t t a c k , r a d
%
a n g l e of a t t a c k f o r s t a l l , r a d
6"
rotor s i d e s l i p angle, rad
Y
b l a d e Lock number, pacR4/Ig
6
equivalent r o t o r blade p r o f i l e drag coefficient
6a
l a t e r a l c y c l i c s t i c k movement, p o s i t i v e t o r i g h t , c m ( i n . )
P
4.
l o n g i t u d i n a l , l a t e r a l , and v e r t i c a l v e l o c i t i e s r e l a t i v e t o t h e a i r mass i n
t h e a i r mass i n t h e hub-body system of a x e s , m/sec ( f t / s e c )
&C
c o l l e c t i v e c o n t r o l i n p u t , p o s i t i v e up, c m ( i n . )
l o n g i t u d i n a l c y c l i c s t i c k movement, p o s i t i v e a f t , c m ( i n . )
6P
p e d a l movement, p o s i t i v e r i g h t , c m ( i n . )
E
hinge o f f s e t r a t i o , e / R
n
stabilizing surface l i f t efficiency factor
11
a x e s system,
0
Euler p i t c h angle, rad
00
blade root c o l l e c t i v e p i t c h , rad
et
t o t a l b l a d e t w i s t ( r o o t minus t i p i n c i d e n c e ) , r a d
A
inflow r a t i o
A
sweepback a n g l e of f i n , r a d
CT
WH
=
A
2(v2
/uH2
r o t o r advance r a t i o ,
+
A2)1/2
+ VH2
aR
P
a i r d e s n t i y , kg/m3 ( s l u g s / f t 3 >
(5
rotor solidity r a t i o , blade area/disk area
T
t i m e constant
9
Euler r o l l angle, rad
R
r o t o r angular v e l o c i t y , rad/sec
Sub s c r i p t s :
a x e s system r e l a t i v e t o a i r mass
B
body-c.g.
c.g.
c e n t e r of g r a v i t y
E
engine
f
fuselage
F
vertical fin
H
hub-body a x e s s y s t e m , hub l o c a t i o n
HS
horizon t a l s t a b i l i z e r
i
induced
MR
main r o t o r
P
p i l o t input
t
throttle
TR
t a i l rotor
W
hub-wind system of a x e s
12
APPENDIX B
AXIS SYSTEMS
Hub Wind
The hub-wind a x i s system i s used i n t h e c a l c u l a t i o n of r o t o r f o r c e s and moments.
The o r i g i n of t h e system i s t h e r o t o r hub, and t h e T ( t h r u s t ) a x i s i s a l i g n e d w i t h
t h e r o t o r s h a f t . The Hw ( h o r i z o n t a l ) a x i s is a l i g n e d w i t h t h e component of relat i v e wind normal t o t h e s h a f t a x i s , and t h e Yw ( s i d e f o r c e ) a x i s completes the
right-handed o r t h o g o n a l set. T h i s a x i s system i s shown i n f i g u r e B-1 a l o n g w i t h t h e
components of the r e l a t i v e wind.
ROTOR SHAFT
OR1ENTATION
(b) PLAN VIEW NORMAL TO
UH, V H PLANE
(a) PERSPECTIVE VIEW
F i g u r e B-1.-
The hub-wind a x i s system w i t h main r o t o r f o r c e , moment,
and v e l o c i t y components def ined.
Hub Body
The hub-body system c o i n c i d e s w i t h t h e hub-wind system when t h e s i d e s l i p a n g l e
i s z e r o . Thus, t h e T a x i s i s a l i g n e d w i t h t h e s h a f t a x i s , and t h e f o r c e HR
l i e s i n t h e X B - Z B p l a n e ( s e e f i g . B-2).
Bw
Body Center of G r a v i t y
All f o r c e s and moments are expressed r e l a t i v e t o t h e body-c.g. system f o r u s e i n
t h e six-degrees-of-freedom r i g i d body e q u a t i o n s of motion. This a x i s system h a s i t s
o r i g i n a t t h e c e n t e r of g r a v i t y w i t h the x a x i s a l i g n e d w i t h t h e l o n g i t u d i n a l a x i s
of t h e h e l i c o p t e r , and t h e z a x i s l y i n g on t h e p l a n e of symmetry ( s e e f i g . B-2).
13
HUB-BODY
'EM
.
AIRCRAFT REFERENCE
SYSTEM
F i g u r e B-2.-
Hub-body,
a i r c r a f t r e f e r e n c e and body-c.g.
a x i s systems.
A i r c r a f t Reference
The a i r c r a f t r e f e r e n c e axes a r e used t o l o c a t e a l l f o r c e and moment g e n e r a t i n g
components and t h e c e n t e r of g r a v i t y . The a i r c r a f t r e f e r e n c e a x e s a r e p a r a l l e l t o
t h e body-c.g. axes. The a x i s o r i g i n i s t y p i c a l l y l o c a t e d ahead and below t h e a i r c r a f t a t some a r b i t r a r y p o i n t w i t h i n t h e p l a n e of s y m m e t r y . S t a t i o n s are measured
p o s i t i v e a f t a l o n g t h e l o n g i t u d i n a l a x i s . B u t t l i n e s are l a t e r a l d i s t a n c e s , p o s i t i v e
t o t h e p i l o t ' s r i g h t . W a t e r l i n e s are measured v e r t i c a l l y , p o s i t i v e upward ( s e e
f i g . B-2).
Local Wind
The local-wind a x e s systems a r e employed f o r c a l c u l a t i o n of l i f t and d r a g
f o r c e s on t h e empennage and on t h e f u s e l a g e . For each empennage s u r f a c e , t h e o r i g i n
i s a t t h e q u a r t e r p o i n t of t h e mean aerodynamic chord, and t h e l i f t f o r c e i s normal
t o t h e r e l a t i v e wind and t o a spanwise l i n e p a s s i n g through t h e q u a r t e r chord p o i n t .
.
14
APPENDIX C
MAIN ROTOR FLAPPING DYNAMICS AND FORCE AND MOMENT CALCULATION
The d e r i v a t i o n of t h e t i p - p a t h p l a n e dynamic e q u a t i o n is g i v e n i n d e t a i l i n
r e f e r e n c e 2 . For t h e n o n t e e t e r i n g N-bladed r o t o r , t h e t i p - p a t h p l a n e dynamic equat i o n s are as f o l l o w s :
15
n
U
3
m
+
+
W
*
I
rllN
I
* 10
W
21-
w
3
?I*
=I*
ffl
I
N
m
0
I
-
m
I
N
uXlc:
@
I
w
Nlm
I
0
rl
+
0
m
+
3
N
I
C
m
+
[I)
W
0
u
wXlG
I
I
rl
I-
W
N1l*
Fl
+
v)
0
n
"w
u
c
(N
W
n
3
1
:
+
+
1w
II
+
8 2
+
nll
t!=l
+
:ai
I
G
C
II
II
ta
tY
I
I
N
N
C
II
tw
16
I
where
P2 =
1
+- KB +-+IBn2
yK1
g-%
8
and
7
For a two-bladed t e e t e r i n g r o t o r , t h e t i p - p a t h p l a n e r e p r e s e n t a t i o n l o s e s i t s p h y s i c a l
meaning. However, i f t h e approximation f o r b l a d e f l a p p i n g i s employed, t h a t is,
$(t) = ao(t)
I*
-
al(t)cos
+ - bl(t)sin +
then a.
i s t r e a t e d as a p r e s e t c o n s t a n t . The c o e f f i c i e n t s a l ( t ) and b , ( t )
t h e n b e s o l v e d f o r by s e t t i n g E = d o = S o = 0 i n e q u a t i o n ( 1 ) .
can
The f o r c e and moment e x p r e s s i o n s , w i t h t h e harmonic terms dropped t h a t correspond
t o t h e number of r o t o r b l a d e s are given below. A d e r i v a t i o n of t h e s e e q u a t i o n s i s
given i n r e f e r e n c e 2.
*MR = nb
2 pacR(QR)2
-1
2
(1
-
1 1
4
(1 -
u2
[++ 5 (1 - + [++ 4 (1 - a. [$+ (1 - E)]K~ + a, [; e ( l 2
E~)X
+ 8,
(1 - E ~ ) ( -B K~ b ~ )
61
+[
k
n
+
+1
4
(1
-
E)]
(?
c o s Bw
17
+ qH
0,
sin B ) ]
W
"bMB ..
-aO
g
E2)]
a
-
(i- );
c
18
-
7
- (1
4
-
~
-
7 a (1
-
4
1
)
E’)(-
~
a
y
PH s i n
~
6
w
n
+
(% -
R
19
a,)]
Q = nb p a c R 2 (RR)
(-&
[l
+
(1
-
- (eo
-
Klao)
[$+ (5 - )'
4
- K 1a 1)
- 4
u a. + bl
-6
16 (1 -
E2)p2
+
+ (- PH
R s i n 6w
qH
+n cos
H
(')
Q +6
.
Q
[(i
- i )(> +
b,)
:
.)1
-e
For t h e c a s e of a two-bladed t e e t e r i n g r o t o r , t h e f o r c e s and moments c a n b e
..
o b t a i n e d by s e t t i n g E = 0 , io = a. = 0 i n t h e above e q u a t i o n s .
The r o t o r p r o f i l e d r a g c o e f f i c i e n t 6 i s r e q u i r e d i n t h e computation of Hw and
An e x p r e s s i o n f o r t h i s c o e f f i c i e n t w a s employed which p r o v i d e s a r e a s o n a b l e match
Q.
w i t h measured s e c t i o n c h a r a c t e r i s t i c s a s follows:
6
= 0.009
+ 0.3
20
(2s
.
where (6CT/aa) i s approximately e q u a l t o an averaged e q u i v a l e n t r o t o r b l a d e a n g l e
of attack.
The r o t o r i n f l o w r a t i o , d e f i n e d as
= -WH
-
cT
nR
Z(p2
+ X2)lj2
i s r e q u i r e d i n t h e computation of t h r u s t . T h i s i m p l i c i t r e l a t i o n s h i p i s s o l v e d i n
t h e computer program through a Newton-Ralphson i t e r a t i v e technique.
2
"
The main rotor-induced v e l o c i t y a t t h e d i s k i s r e q u i r e d i n subsequent c a l c u l a t i o n s of t h e r o t o r - i n d u c e d v e l o c i t i e s on t h e h e l i c o p t e r components.
The e x p r e s s i o n
f o r t h i s v e l o c i t y i s as follows:
Transformation
T,
The c a l c u l a t i o n of t h e r o t o r f l a p p i n g dynamics and t h e r o t o r f o r c e s and moments
r e q u i r e s t h e a n g u l a r v e l o c i t i e s and a c c e l e r a t i o n s e x p r e s s e d i n t h e hub-body system of
a x e s and t h e a n g l e of s i d e s l i p a t t h e hub. I n a d d i t i o n , the c y c l i c p i t c h must b e
e x p r e s s e d i n t h e hub-wind system. The r e q u i r e d v e l o c i t i e s and a c c e l e r a t i o n s are
o b t a i n e d from t h o s e e x p r e s s e d i n t h e body-c.g. system as f o l l o w s :
pH = p c o s is + r s i n is
6
6,
=
u
= tuB
H
+
c o s is
-
+C
s i n is
rB(BL
)
c.g.
-
qB(WLH
-
WL
c.g.
+ P B ( ~ ~ ~ -. qB(sTAc,g.
~ , )
-
>]cos i
S
ST%)]sin i
S
'E-
-.
where is i s t h e forward t i l t of the s h a f t axis r e l a t i v e t o t h e body-c.g.
Z-axis
( s e e f i g . B-2) and STAY WL, and BL are c o o r d i n a t e s i n t h e a i r c r a f t r e f e r e n c e system
of a x e s . I t i s assumed t h a t t h e r o t o r hub l i e s i n t h e BL = 0 p l a n e .
The s i d e s l i p of t h e r o t o r i s then d e f i n e d as:
21
(B, is d e f i n e d as z e r o if VH = UH = 0.) F i n a l l y t h e c y c l i c p i t c h must b e expressed
i n t h e hub-wind a x i s system. The e x p r e s s i o n s are:
=
Als c o s
Blc =
sin
A
1c
$
-
Bw
+ Bls
W
B l s s i n Bw
Bw
COS
T2
Transformation
The r o t o r f o r c e s and moments c a l c u l a t e d i n t h e hub-wind a x e s system must b e
expressed i n t h e body-c.g. system as i n p u t s t o t h e six-degree-of-freedom rigid-body
e q u a t i o n s of motion. To accomplish this, t h e f o r c e s and moments are f i r s t expressed
i n t h e hub-body s y s t e m of axes and then t r a n f e r r e d t o t h e body-c.g. system.
The f o r c e s and moments i n t h e
t i o n as f o l l o w s :
HH = H
W
YH = -H
p l a n e are modified by t h e f i r s t transforma-
x-y
W
+ Yw
c o s 8,
W
c o s Bw
= -Mw
s i n €3
LH
+ YW
s i n f3
% = Mw
s i n B,
+ Lw
s i n BW
+ LW
W
c o s ,6
cos
Bw
The t r a n s f o r m a t i o n t o t h e body-c.g. system a c c o u n t s f o r t h e s h a f t - t i l t and t h e moment
a b o u t t h e c.g. caused by t h e r o t o r f o r c e s . The f o r c e s e x p r e s s e d i n the body-c.g.
system are:
-
= T sin i
S
Ym
= I’$
-
-
s i n iS
system are:
Zm(STA c . g .
NMR = Q c o s is
S
= YH
‘MR = -T c o s iS
The moments i n t h e body-c.g.
cos i
+ LH
-
ST+)
-
%(mH
s i n is + % R ( ~ ~ g.
c.
22
- WLc . g .
+
YMR(STAc.g
.-
ST+)
APPENDIX D
TAIL ROTOR FLAPPING AND FORCE CALCULATION
The t a i l r o t o r i s assumed t o b e t e e t e r i n g w i t h a c o n s t a n t b u i l t - i n c o n i n g a o .
The a n g u l a r r a t e of t h e t a i l r o t o r i s s u f f i c i e n t l y h i g h so t h a t t h e t i p - p a t h dynamics
i n t h e f l a p p i n g e q u a t i o n s are
may b e ignored. Under t h e s e c o n d i t i o n s , a, and b,
assumed t o b e c o n s t a n t , and a s t e a d y - s t a t e s o l u t i o n of t h e e q u a t i o n s i s o b t a i n e d t o
y i e l d t h e s e q u a n t i t i e s as f o l l o w s :
-.
.
where
4
ATR
=
1
-4TR +
F)(l+ 5
2
K:TR(l
+
PgR)
The f o r c e s on t h e t a i l r o t o r a r e then c a l c u l a t e d as f o l l o w s :
- a
OTR
(i+ *)
KITR
+
21
*.
23
1
+
jJ KlTRblTR
-
+ -blm
-4
a
'TR
3
The p o s i t i v e d i r e c t i o n s f o r t a i l r o t o r t o r q u e and r o t a t i o n are shown on
f i g u r e D-1.
THRUST ALONG SHAFT
INTO PAGE
TR
G
ROTATION
F i g u r e D-1.-
Tail r o t o r f o r c e s and moments.
24
+ - 3A
4
a
TR 1TR
-
a o ~ b l ~ ~
6
+ -' +~ T- R
.
3-
8
ATR
alTRplTR]
3
16 K 1 ~ ~ b l ~2
~ p m 16
+
:i
+
The r o t o r b l a d e p r o f i l e d r a g c o e f f i c i e n t , 6 , i s r e q u i r e d i n t h e computation of
HTR and QTR. T h i s i s c a l c u l a t e d s i m i l a r l y t o t h e c a s e of t h e main r o t o r as f o l l o w s :
'.
The i n f l o w r a t i o , a s i n t h e c a s e of t h e main r o t o r , must b e c a l c u l a t e d from t h e
following relationship:
25
T h i s i m p l i c i t r e l a t i o n s h i p i s s o l v e d through a n i t e r a t i o n procedure.
Main Rotor I n t e r f e r e n c e and Transformation
T,
The c a l c u l a t i o n of the t a i l r o t o r f o r c e s and moments r e q u i r e s t h e l o c a l f l o w
v e l o c i t y components i n t h e hub-wind axes system f o r t h e t a i l r o t o r . The v e l o c i t y a t
t h e t a i l r o t o r i n c l u d e s p r o v i s i o n f o r t h e downwash c o n t r i b u t e d by t h e ' f l o w f i e l d of
t h e main r o t o r . For t h e s e c a l c u l a t i o n s , t h e t a i l r o t o r i s assumed t o l i e i n t h e
x-z plane.
The v e l o c i t i e s a t t h e r o t o r hub i n an a x i s system c o - d i r e c t i o n a l w i t h t h e
body-c.g. system are:
UTR
= u
B
TR = vB
wTR = w
B
+ p B ( m m - WLc . g . 1 -
rB(STATR - STA
c.g.
+
+ wiTR
qB(STATR - STA
)
c.g.
i s t h e downwash v e l o c i t y due t o t h e main r o t o r . The downwash
The q u a n t i t y w i
v e l o c i t y is r e p r e s e n t s i n t h e modes as a f u n c t i o n of t h e wake a n g l e x and s i d e s l i p .
The v a r i a t i o n of t h e downwash v a r i e s n o n l i n e a r l y w i t h l o c a t i o n i n t h e wake. T h e r e f o r e ,
t h e downwash must be determined uniquely f o r each l o c a t i o n from d a t a such as t h o s e
p u b l i s h e d i n r e f e r e n c e 10. The r e p r e s e n t a t i o n i n t h e model c o n s i s t s of t h e downwash
e x p r e s s e d as a f u n c t i o n of a power series i n t h e wake a n g l e a t a n g l e s of s i d e s l i p of
0", ?goo, and 180" a s f o l l o w s :
where v i i s t h e momentum-theory v a l u e of t h e r o t o r - i n d u c e d v e l o c i t y . The c o o r d i n a t e
system employed i n r e f e r e n c e 10 i s c e n t e r e d a t t h e hub, and t h e X and Y a x e s l i e i n
t h e t i p - p a t h plane. Accordingly, t h e t a i l r o t o r l o c a t i o n i n t h i s c o o r d i n a t e system
varies w i t h t h e v a l u e of a l , t h e t i l t of t h e t i p - p a t h p l a n e . I n view of t h e a p p r o x i mate n a t u r e of t h e wake estimate, which c a n v a r y widely w i t h d i s k l o a d d i s t r i b u t i o n ,
i n c l u s i o n of t h i s level of d e t a i l i s unwarranted. Accordingly, t h e downwash i s e s t i mated f o r a c o n s t a n t l o c a t i o n r e f e r e n c e d t o t h e hub-body system and t h e wake a n g l e
r e f e r r e d t o t h a t s y s t e m . Hence,
I n t e r p o l a t i o n f o r t h e v a l u e of t h e downwash a t a r b i t r a r y s i d e s l i p a n g l e s i s accomp l i s h e d as i n t h e f o l l o w i n g example, where B h a s been determined t o b e between 290":
S i m i l a r l y , when 90" < 6
270", t h e downwash i s c a l c u l a t e d as f o l l o w s :
26
--*
W iTR
=
[(?)
i
lcos B I
@=le0
+
(%)
f3=9 0
Isin BI]Vi
The advance r a t i o f o r t h e t a i l r o t o r , which is assumed t o b e i n the
i s then:
.
x-z
plane,
and t h e r o t o r s i d e s l i p a n g l e i s d e f i n e d f o r t h e t a i l r o t o r as:
Then, i n t h e hub-wind a x i s system d e f i n e d f o r t h e t a i l r o t o r , t h e a n g u l a r v e l o c i t i e s
are :
Transformation
T,
The t r a n s f o r m a t i o n t o e x p r e s s t h e t a i l r o t o r f o r c e s and moments i n t h e body-c.g.
system of a x e s i s as f o l l o w s :
XTR = - ( Y T R ) ~s i n BTR
-
( H T R ) ~c o s BTR
-
If-
27
APPENDIX E
EMPENNAGE FORCES AND MOMENTS
The e x p r e s s i o n s f o r t h e l i f t and d r a g c o e f f i c i e n t s f o r t h e v e r t i c a l f i n and t h e
h o r i z o n t a l s t a b i l i z e r are i d e n t i c a l when r e f e r e n c e d t o a l o c a l wind-axes system f o r
the p a r t i c u l a r surface.
(See appendix B.)
The c o n d i t i o n s a t t h e s t a l l of each s u r f a c e a r e d e f i n e d as f o l l o w s :
as
cLM
--
as =
a s 1.
a
IT
4
IT
IT
4
as
a1 = 1.2as
as
< -71
4
-
The a n g l e of a t t a c k f o r i n p u t i n t o t h e e x p r e s s i o n s f o r t h e l i f t and d r a g c o e f f i c i e n t
is:
where
ai
IT
ai = a
0 5 a < -
ai = -a
- - 2- I a < o
ai=71+a
-TI
2
IT
represents t h e angle-of-attack
I a < - -IT
2
i n p u t t o t h e l i f t and d r a g e q u a t i o n s .
The l i f t and d r a g c o e f f i c i e n t s a r e c a l c u l a t e d u s i n g t h e f o l l o w i n g e x p r e s s i o n s :
0 5 a i < as
cLo = aai
cDP
= -0.1254
+ 0.9415ai
+ 0.977525 s i n 2 a i
28
IT
0.35 < a i < 7
where
C D ~ is t h e p r o f i l e d r a g c o e f f i c i e n t .
c;
cD = ‘DP + 0.8rAR
Depending on t h e q u a d r a n t i n which t h e a n g l e of a t t a c k l i e s , t h e l i f t c o e f f i c i e n t
f o r t h e s u r f a c e is:
n
2
O Z a < -
CL = CL,
- - ?T<
CL = -CLo
2 -
CL = -0.8CL
CL
=
IT
0
2
a<IT
IT
- . r r _ < a < -71-
0.8C~
2
a
Horizontal T a i l
The l i f t c u r v e s l o p e of t h e h o r i z o n t a l t a i l i s c a l c u l a t e d from elementary l i f t i n g l i n e t h e o r y and i s c o r r e c t e d f o r s i d e s l i p a s f o l l o w s :
Main r o t o r i n t e r f e r e n c e and t r a n s f o r m a t i o n T5- The v e l o c i t y a t the h o r i z o n t a l
This f a c t o r i s c a l c u t a i l i n c l u d e s a f a c t o r t o a c c o u n t f o r r o t o r downwash, W i H T .
l a t e d s i m i l a r l y t o t h a t calculated f o r the t a i l rotor. I n calculating the v e l o c i t i e s
a t t h e h o r i z o n t a l t a i l , t h e r e l a t i v e l y small c o n t r i b u t i o n of r o l l rate i s ignored.
The e x p r e s s i o n s f o r t h e v e l o c i t y components a t t h e h o r i z o n t a l t a i l are as f o l l o w s :
The a n g l e of a t t a c k and a n g l e of s i d e s l i p of t h e h o r i z o n t a l t a i l are then:
’H S = sin-’
cz)
where
29
The a n g l e iHsi s t h e i n c i d e n c e of t h e h o r i z o n t a l t a i l . Note t h a t t h e e f f e c t of a
cambered s u r f a c e may b e accounted f o r by s e l e c t i n g t h i s parameter t o d i f f e r from t h e
a n g l e of t h e chord p l a n e by a n amount e q u a l t o t h e a n g l e of attack f o r z e r o l i f t .
The f a c t o r employed t o c a l c u l a t e l i f t and d r a g f o r c e s from t h e l i f t and d r a g
c o e f f i c i e n t s is:
where t h e c o n s t a n t ~ H S i s used t o account f o r f u s e l a g e blockage o r t h e h o r i z o n t a l
t a i l area c o n s i d e r e d t o b e w i t h i n t h e f u s e l a g e .
Transformation
h o r i z o n t a l t a i l are:
T6-
The f o r c e s i n t h e body-c.g.
system of a x e s a t t r i b u t e d t o t h e
The f o r c e s a r e assumed t o a c t a t t h e q u a r t e r chord p o i n t of t h e h o r i z o n t a l t a i l
a t a spanwise l o c a t i o n c o r r e s p o n d i n g t o t h e c e n t e r of area. The moments are:
Vertical F i n
The estimate of t h e l i f t c u r v e s l o p e of t h e v e r t i c a l f i n i n c l u d e s t h e e f f e c t of
sweep a s w e l l a s of s i d e s l i p as f o l l o w s :
Transformation T7- The computation of t h e v e l o c i t y a t t h e v e r t i c a l f i n i g n o r e s
t h e s m a l l c o n t r i b u t i o n from r o l l r a t e and i n c l u d e s a t e r m i n t h e sidewash due t o t h e
induced v e l o c i t y of t h e t a i l r o t o r . The t h r e e components of v e l o c i t y a r e as f o l l o w s :
30
The c o n s t a n t kvTR can b e a d j u s t e d t o a c c o u n t f o r t h e f r a c t i o n of t h e v e r t i c a l t a i l
is computed
immersed i n t h e t a i l r o t o r wake. The main r o t o r i n t e r f e r e n c e f a c t o r wi
F
s i m i l a r l y t o t h e f a c t o r f o r downwash a t t h e t a i l r o t o r .
The a n g l e s of a t t a c k and s i d e s l i p of t h e v e r t i c a l f i n i n t h e l o c a l wind c o o r d i n a t e system are:
I
.where
As i n t h e c a s e of t h e h o r i z o n t a l t a i l , t h e a n g l e
camber of t h e v e r t i c a l f i n .
31
iF can b e a d j u s t e d t o a c c o u n t f o r
APPENDIX F
CALCULATION OF FUSELAGE FORCES AND MOME"l'S
I n c a l c u l a t i n g the f u s e l a g e f o r c e s and moments, i t i s assumed that t h e l o n g i t u d i n a l f o r c e s and moments are dependent on a n g l e of a t t a c k and t h e l a t e r a l f o r c e s and
moments are dependent on a n g l e of s i d e s l i p . The e x c e p t i o n i s t h e d r a g f o r c e , which
i s assumed t o have a c o n t r i b u t i o n from b o t h a n g l e of a t t a c k and s i d e s l i p . T h i s repr e s e n t a t i o n i s a g r o s s s i m p l i f i c a t i o n , and i f d e t a i l e d test d a t a are a v a i l a b l e on
which t o b a s e a more s o p h i s t i c a t e d r e p r e s e n t a t i o n , t h e y should be s u b s t i t u t e d .
(See,
e.g., r e f . 8 . )
Low-angle approximation
D = D,
I_
a,B = -15" t o 15"
+ DB
DB
(
a2D/q
= qf
aB2
B2)
High-angle approximation
)
s i n B cos B
s i n a1 s i n 2 ,
M
= qf
D = D,
1
)
sin B
(M,=90~ I s i n a I s i n a)
N = qf
+ DO
(
~
~
I sin BI
s i0 n B)
BI
s i n 6')
=
~
DB = q f ( D g = g o " I s i n
~
Phasing between t h e low-angle approximation and t h e high-angle approximation i s
based on a complex phase a n g l e and l i n e a r i n t e r p o l a t i o n . The complex phase a n g l e i s
g i v e n by :
L i n e a r i n t e r p o l a t i o n based on t h i s phase a n g l e i s employed o v e r t h e r a n g e of
15" < ITFUSI
30".
32
-.
Main r o t o r i n t e r f e r e n c e and t r a n s f o r m a t i o n Tg- The f u s e l a g e a n g l e s of a t t a c k ,
a n g l e of s i d e s l i p , and dynamic p r e s s u r e i n c l u d e a n averaged e f f e c t of r o t o r downwash.
The e x p r e s s i o n f o r downwash w a s o b t a i n e d f o r t y p i c a l s i n g l e r o t o r h e l i c o p t e r s from
a n e m p i r i c a l f i t t o d a t a p r e s e n t e d i n r e f e r e n c e 10. The v e l o c i t i e s on t h e f u s e l a g e
are as f o l l o w s :
wf =
.Where wif
WB
+ w if
i s t h e induced v e l o c i t y due t o t h e r o t o r and i s d e f i n e d as:
(F)
= 1.299
The r o t o r wake a n g l e
x
+ 0 . 6 7 1 ~-
1 . 1 7 2 ~+~0 . 3 5 1 ~ ~
i s d e f i n e d as:
The a n g l e s of a t t a c k and s i d e s l i p a r e then:
af = tan-’
Wf
Uf
where
’-
The dynamic p r e s s u r e on t h e f u s e l a g e i s based on t h e v e l o c i t y , i n c l u d i n g t h e e f f e c t
of r o t o r downwash.
T r a n s f o r m a t i o n T l o - The f u s e l a g e f o r c e s and moments are c a l c u l a t e d i n t h e wind
system of a x e s a b o u t t h e f u s e l a g e r e f e r e n c e p o i n t . The t r a n s f o r m a t i o n must t r a n s f e r
t h e moment t o t h e c e n t e r of g r a v i t y and e x p r e s s a l l f o r c e s and moments i n t h e body
system of axes.
33
The forces and moments i n the body-c.g.
ing equations:
Xf
=
-D
cos
B cos a
Yf = Y cos B
Zf = -L cos a
-
-Y
sin f3 s i n a
+L
system of axes are given by the follow-
sin a
D sin B
-
D cos B s i n a
-
Y sin
sin a
34
RPM GOVERNOR
The rpm governor model p r o v i d e s f o r an rpm d e g r e e of freedom w i t h simple e n g i n e
and governor dynamics. When the rpm governor o p t i o n i s u s e d , t h e main r o t o r and t a i l
r o t o r s p e e d s are changed based on c u r r e n t t o r q u e r e q u i r e m e n t s and e n g i n e power availa b l e . T r i m i n i t i a l c o n d i t i o n s e s t a b l i s h b a s e l i n e v a l u e s of r o t o r speed and e n g i n e
t o r q u e . F l i g h t v a r i a t i o n s from t h a t t r i m c o n d i t i o n r e s u l t i n changing t o r q u e r e q u i r e ments which c a u s e r o t o r speed v a r i a t i o n s which f e e d through t h e governor c o n t r o l l a w s
t o p r o v i d e f u e l flow changes. The f u e l f l o w changes p r o v i d e e n g i n e power changes
which are c o n v e r t e d t o e n g i n e t o r q u e a v a i l a b l e .
c
The rpm d e g r e e of freedom may b e d e s c r i b e d by the e q u a t i o n :
.
where :
QE = e n g i n e t o r q u e
QR = r e q u i r e d r o t o r t o r q u e
J = rotor rotational inertia
h
= r o t o r speed r a t e of change
Fo low f l a p p i n g h i n g e o f f s e t s , t h e r o t t i o n a l i e r t i , J , may b e a p roximated from
t h e b l a d e f l a p p i n g i n e r t i a , IB:
J = NIB
where
N
i s t h e number of b l a d e s .
The main - \ ' t o r t o r q u e r e q u i r e d i s a complicated f u n c t i o n of many v a r i a b l e s
i n c l u d i n g b l a d e p i t c h s e t t i n g s , a i r s p e e d , i n f l o w v e l o c i t y , f l a p p i n g a n g l e s , and r o t o r
speed. The t o r q u e e q u a t i o n p r o v i d e s t h e n e c e s s a r y c a l c u l a t i o n . The r o t o r speed gove r n i n g of t h i s model u s e s l i n e a r c o n t r o l theory based on an o p e r a t i n g p o i n t . Thus,
t h e r o t o r speed, Q , t a k e s t h e form:
R = R,
where
R,
i s t h e trim r o t o r speed and
+
AR
AR
i s t h e rotor-speed v a r i a t i o n .
?*
A d e t a i l e d r e p r e s e n t a t i o n of e n g i n e t o r q u e r e q u i r e s a complicated n o n l i n e a r
f u n c t i o n of many v a r i a b l e s i n c l u d i n g o p e r a t i n g power s e t t i n g , ambient p r e s s u r e and
t e m p e r a t u r e , and f u e l flow, W f .
For t h i s s i m p l i f i e d model, rpm governing acts on
f u e l f l o w t o c o n t r o l e n g i n e t o r q u e . A g a s t u r b i n e e n g i n e produces power which must
b e c o n v e r t e d t o t o r q u e based on t h e c u r r e n t r o t o r speed.
35
I t i s i m p o r t a n t t o n o t e that t h e c u r r e n t r o t o r speed i s used, n o t t h e o p e r a t i n g
p o i n t speed, Q,,. Thus, the power-to-torque c o n v e r s i o n f a c t o r i s always changing.
The e n g i n e power response t o f u e l flow i s simply modeled as a f i r s t o r d e r l a g :
The time c o n s t a n t ,
operating point.
TE,
and g a i n , KE, are s e l e c t e d based on e n g i n e c h a r a c t e r i s t i c s and
The complete rpm d e g r e e of freedom and engine-governor model i s shown i n
f i g u r e 3.
-
Note t h a t a t h r o t t l e on/off s w i t c h h a s been added t o a l l o w t u r n i n g t h e power on
o r o f f . A s e p a r a t e time c o n s t a n t f o r t h e t h r o t t l e i s provided t o a l l o w f o r a l a r g e
power change time c o n s t a n t f o r t h e t h r o t t l e and a small power change t i m e c o n s t a n t
f o r normal engine o p e r a t i o n .
The governor c o n t r o l l a w is given by:
AWE = -@g1
+
which p r o v i d e s p r o p o r t i o n a l , i n t e g r a l , and r a t e feedback.
-s
.
36
APPENDIX H
COCKPIT CONTROLS AND CYCLIC CONTROL PHASING
Y
C y c l i c , c o l l e c t i v e , and p e d a l c o n t r o l s may b e s p e c i f i e d from a z e r o p o s i t i o n
e i t h e r c e n t e r e d o r f u l l l e f t o r down. The c y c l i c c o n t r o l p o s i t i o n must b e c o n s i s t e n t
between l o n g i t u d i n a l and l a t e r a l c y c l i c , t h a t i s , c e n t e r e d o r f u l l l e f t and forward.
The z e r o p o s i t i o n of t h e c o l l e c t i v e o r p e d a l s may b e s p e c i f i e d independently. The
t a b l e below l i s t s t h e z e r o p o s i t i o n , p o s i t i v e d i r e c t i o n , and s i g n of t h e moment o r
f o r c e produced i n t h e body system of axes. A l t e r n a t e c o n t r o l z e r o p o s i t i o n convent i o n s are l i s t e d i n p a r e n t h e s e s .
Control
Zero p o s i t i o n
Longitudinal cyclic,
Centered
( f u l l forward)
Aft
+M
Lateral c y c l i c , 6,
Centered
( f u l l left)
Right
+L
C o l l e c t i v e , tic
F u l l down
UP
-Z
P e d a l s , 6p
Centered
(full left)
Right p e d a l
forward
+N
6,
Positive direction
Moment/force
The c o n t r o l g e a r i n g , r i g g i n g , and c y c l i c c o n t r o l phasing are governed by t h e following equations:
e,
'OTR
T h e terms
24
c
, C,,
+ c,
= C,6,
= C b
8
and C,
P
+c,
are t h e c o n s t a n t o r r i g g i n g terms f o r each
c o n t r o l . The terms C, through C,
can b e used t o a d j u s t f o r t h e phase a n g l e between
t h e c y c l i c c o n t r o l i n p u t and t h e r e s u l t i n g f l a p p i n g . These phase a n g l e terms may b e
s p e c i f i e d i n two ways. F i r s t , a c y c l i c c o n t r o l phase r e l a t i o n s h i p may b e s p e c i f i e d
on t h e b a s i s of main r o t o r dynamic p r o p e r t i e s :
c,
=
1
-
( 8 / 3 ) ~+ 2~~
(CK2)
1- (4/3)€
37
c,
= ‘C,
(s)
The terms CK1 and CK2 a r e t h e l o n g i t u d i n a l and l a t e r a l c y c l i c c o n t r o l s e n s i t i v i t i e s ,
r e s p e c t i v e l y . As an a l t e r n a t i v e , a c o n t r o l phase a n g l e , J l o , may be s p e c i f i e d , which
provides the following r e l a tionships:
C, = (CK2)cos
q0
Y
.
C, = (CK2)sin Jl,,
c,
=
-c,
(s)
.
38
APPENDIX I
LINEARIZED SIX-DEGREE-OF-FREEDOM
REPRESENTATION OF HELICOPTER DYNAMICS
The linear , f i r s t - o r d e r set of d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g the r i g i d body
motion of t h e h e l i c o p t e r are of the form:
2
yr
= [FIX
+
[GI6
r e p r e s e n t s the p e r t u r b a t i o n s from t r i m of the s t a t e variables UB, WB, qBy
and r B ; and 6 r e p r e s e n t s t h e d e v i a t i o n s from t h e t r i m c o n t r o l p o s i t i o n s of 6,, 6,, 6,, and 6p. The l i n e a r r e p r e s e n t a t i o n i s v a l i d o n l y i f the i n i t i a l
a n g u l a r v e l o c i t i e s p ~ qBy
,
and r B are z e r o .
where
x
0 , VB, PB, $I,
=-
The elements of t h e F and G matrices are of two types. The f i r s t t y p e c o n s i s t s
of i n e r t i a l and g r a v i t a t i o n a l terms t h a t can b e o b t a i n e d a n a l y t i c a l l y from t h e equat i o n s of motion. The second t y p e c o n s i s t s of p a r t i a l d e r i v a t i v e s a r i s i n g from aerodynamic f o r c e s and moments. The f o r c e and moment d e r i v a t i v e s are o b t a i n e d by conside r i n g b o t h p o s i t i o n and n e g a t i v e p e r t u r b a t i o n s from t r i m . For example:
The elements of t h e
i n t a b l e 1-1.
F and G
m a t r i c e s and t h e s t a t e v a r i a b l e v e c t o r s are given
39
I
I
I
I
I
I
I
I
I
a
I
I
I
I
I
_-
I
8
a
I
I
I
I
I
7--r--I
I
I
I
I
I
I
I
m
a
a
ma
a
0
a
m
0.
a
E+
n
a
W
4
m
a
W
h
N
N X
3"
a
m
a
Y
U
W
I
a
"
N
H
X
W
H
v
4
v
II
II
II
x
w
H
v
U
40
APPENDIX J
CONFIGURATION D E S C R I P T I O N REQUIREMENTS
Table J-1 l i s t s t h e p a r a m e t e r s r e q u i r e d t o d e s c r i b e a h e l i c o p t e r c o n f i g u r a t i o n
f o r u s e i n t h e computer s i m u l a t i o n . L i s t e d are t h e parameter name, a l g e b r a i c symbol,
computer mnemonic, and u n i t s f o r each parameter. I n a d d i t i o n , a n example v a l u e based
on a n AH-1G i s provided f o r each parameter. AH-1G p a r a m e t e r s are based on v a l u e s i n
r e f e r e n c e 8.
8
.
c
2-
41
TABLE J-1.- CONFIGURATIONDESCRIPTION REQUIREMENTS
Name
Algebraic
symbol
CmPuter
mnemonic
Units
Example
value
Main r o t o r (MR) group
MR r o t o r r a d i u s
RMR
MR chord
ROTOR
f t
CHORD
ft
22
2.25
32.88
MR r o t a t i o n a l speed
QMR
OMEGA
rad/sec
Number of b l a d e s
nb
BLADES
N-D
2
MR Lock number
YMR
GAMMA
N-D
5.216
EPSLN
percent/100
0
MR hinge o f f s e t
E:
MR f l a p p i n g s p r i n g c o n s t a n t
KB
AKBETA
lb-f t / r a d
0
MR p i t c h - f l a p coupling t a n g e n t of
Kl
AKONE
N-D
0
m
-
63
‘~MR
a
THETT
rad
-0.17453
AOP
rad
0.048
MR s o l i d i t y
UMR
SIGMA
N-D
0.0651
MR l i f t curve s l o p e
aMR
ASLOPE
rad’’
6.28
CTM
N-D
0.165
MR b l a d e t w i s t
MR precone a n g l e ( r e q u i r e d f o r
teetering rotor)
MR maximum t h r u s t
OMR
‘Tmax
MR l o n g i t u d i n a l s h a f t t i l t ( p o s i t i v e
forward)
iS
CIS
rad
0
MR hub s t a t i o n l i n e
STAH
STAH
in.
200
MR hub w a t e r l i n e
WLH
WLH
in.
152.76
TR r a d i u s
RTR
RTR
f t
TR r o t a t i o n a l speed
QTR
OMTR
r a d 1sec
TR Lock number
YTR
GAMATR
N-D
2.2337
TR s o l i d i t y
OTR
STR
N-D
0.105
FKITR
N-D
0.5773
AOTR
rad
0.02618
THETR
rad
0
ATR
rad’’
6.28
TR hub s t a t i o n l i n e
a~~
STAm
STATR
in.
520.7
TR hub w a t e r l i n e
WLTR
WLTR
in.
118.27
T a i l r o t o r (TR) group
TR p i t c h - f l a p coupling t a n g e n t of
K
1
~
~
4.25
168.44
63
TR precone
TR b l a d e t w i s t
TR l i f t curve s l o p e
a
OTR
‘~TR
42
-9
TABLE J-1.
-
Continued
A1geb ra i c
symbol
Name
.
C ompu t e r
mnemonic
Units
Examp l e
value
A i r c r a f t mass and i n e r t i a
A i r c r a f t weight
WAITIC
lb
8000
A i r c r a f t roll i n e r t i a
XIXXIC
slug-f t
2700
Aircraft pitch inertia
XIYYIC
slug-f t
12800
A i r c r a f t yaw i n e r t i a
XIZZIC
slug-f t
10800
A i r c r a f t c r o s s product of i n e r t i a
XIXZIC
slug-f t
950
C e n t e r of g r a v i t y s t a t i o n l i n e
STACG
in.
196
C e n t e r of g r a v i t y w a t e r l i n e
WLCG
in.
73
C e n t e r of g r a v i t y b u t t l i n e
BLCG
in.
0
Fus aerodynamic r e f e r e n c e p o i n t
s t a ti o n l i n e
STAACF
in.
200
Fus aerodynamic r e f e r e n c e p o i n t
w a t e r 1i n e
WLACF
in.
54
D1
f t2
Fuselage (Fus)
Fus d r a g ,
c1 =
B = 0
5.5
Fus d r a g , v a r i a t i o n w i t h
ci
D2
f t2/rad
-4.01
Fus d r a g , v a r i a t i o n w i t h
a2
D3
f t /rad2
41.56
Fus d r a g , v a r i a t i o n w i t h
B2
D4
f t2/rad2
141.16
Fus d r a g , cx = 90"
D5
f t2
84.7
Fus d r a g , 6 = 90"
D6
f t2
156.1
XLO
ft2
-4.11
XL1
f t2/rad
15.64
Y1
f t2/rad
93.85
YL1
ft3/rad
246.31
YL2
ft3
Fus l i f t , a =
B
LO
-
= 0
Fus l i f t , v a r i a t i o n w i t h
q
a (L/q)
a
Fus s i d e f o r c e , v a r i a t i o n w i t h
aa
B
Fus r o l l i n g moment, v a r i a t i o n w i t h
B
a (Y/q)
a6
a(R/q)
at3
Fus r o l l i n g moment, B = 90"
43
0
TABLE J-1.- Continued.
Algebraic
symbol
Name
M
Fus p i t c h moment, a = B = 0
9
a
Fus p i t c h moment, v a r i a t i o n w i t h
aa
MI
Fus p i t c h moment, a = 90”
Fus yaw moment, v a r i a t i o n w i t h
a (M/q)
4 a=90
f3
a (N/q)
Fus yaw moment, B = 90”
Computer
mnemonic
Units
Example
value
-6.901
XM1
ftj
xM2
f t 3/ r a d
280.405
xM3
ft3
300
m
xN1
f t 3/ r a d
-913.35
xN2
ft3
-600
H o r i z o n t a l s t a b i l i z e r (HS)
HS s t a t i o n
STAHS
STAHS
in.
398.5
HS w a t e r l i n e
WLHS
WLHS
in.
56.0
HS i n c i d e n c e a n g l e
iHS
AIHS
rad
0
HS area
sHS
SHS
ft2
14.7
HS aspect r a t i o
ARHS
ARHS
N-D
3.0
HS maximum l i f t curve s l o p e
C
CLMHS
N-D
1.2
X”
N-D
0.8
XKVMR
N-D
1.0
ImaxHS
HS dynamic p r e s s u r e ratio
“HS
Main r o t o r induced v e l o c i t y e f f e c t
a t HS
K~~~
Vertical f i n (VF)
VF s ta t i o n l i n e
STAVF
STAVF
in.
50 1
Vertical f i n waterline
WLVF
WLVF
in.
84
VF I n c i d e n c e a n g l e
iVF
AIFF
rad
0
VF area
SVF
SF
ft2
18.6
VF aspect r a t i o
ARVF
AFtF
N-D
1.56
VF sweep a n g l e
AF
C
ImaxVF
ALMF
rad
0.7853
CLMF
N-D
1.2
VF dynamic p r e s s u r e r a t i o
qVF
VNF
N-D
0.9
T a i l r o t o r induced v e l o c i t y e f f e c t
a t VF
k~~~
XKVTR
N-D
0.9
VF maximum l i f t curve s l o p e
44
-?
s
c
TABLE J-1.-
Name
Concluded.
Algebraic
symbol
Computer
mnemonic
Units
Example
value
Controls
&
Swashplate l a t e r a l c y c l i c p i t c h f o r
zero lateral cyclic s t i c k
C
CAI S
rad
0.0
Swashplate l o n g i t u d i n a l c y c l i c p i t c h
f o r zero longitudinal cyclic s t i c k
CB1
CBIS
rad
0.0
Longitudinal c y c l i c control
sensitivity
CKl
CK1
rad/in.
0.03927
Lateral c y c l i c c o n t r o l s e n s i t i v i t y
CK,
CK2
rad/in.
0.02618
Main r o t o r r o o t c o l l e c t i v e p i t c h f o r
zero c o l l e c t i v e s t i c k
C,
c5
rad
0.1501
Main r o t o r c o l l e c t i v e c o n t r o l
sensitivity
C6
C6
rad/in.
0.036652
T a i l rotor root collective pitch for
zero pedal position
C,
c7
rad
0.11781
Pedal s e n s i t i v i t y
‘8
C8
rad/in.
0.08055
A
~
~
S
c
b
+
45
REFERENCES
l . 1 .
2.
' 3.
4.
S i n a c o r i , J. B.; S t a p l e f o r d , R. L.; Jewell, W. F.; and Lehman, J. M.:
R e s e a r c h e r ' s Guide t o the NASA Ames F l i g h t S i m u l a t o r f o r Advanced A i r c r a f t
(FSAA), NASA CR-2875, Aug. 1977.
Chen, R. T. N.:
A S i m p l i f i e d Rotor System Mathematical Model f a r P i l o t e d F l i g h t
Dynamics S i m u l a t i o n , NASA TM-78575, May 1979.
Chen, R. T. N.:
E f f e c t s of Primary Rotor P a r a m e t e r s on F l a p p i n g Dynamics, NASA
TP-1431, Jan. 1980.
Y
Gessow, A.; and Meyers, J r . , G. D.:
Ungar Pub. Co. (New York), 1952.
A
Aerodynamics of t h e H e l i c o p t e r .
-
Frederick
L
and C u r t i s s , J r . , H. C.:
Aerodynamic C h a r a c t e r i s t i c s of H e l i c o p t e r
Dept. of Aerospace and Mechanical Eng. Rep. 659, P r i n c e t o n Univ.,
5.
Seckel, E.;
Rotors.
196 2.
6.
White, F.; and Blake, B. B.:
Improved Method of P r e d i c t i n g H e l i c o p t e r C o n t r o l
Response and Gust S e n s i t i v i t y . Paper 79-25, 3 5 t h Ann. N a t . Forum of t h e AHS,
Wash., D.C., May 1979.
7.
Chen, R. T. N.:
S e l e c t i o n of Some Rotor P a r a m e t e r s t o Reduce P i t c h - R o l l Coupling
of H e l i c o p t e r F l i g h t Dynamics. P r e p r i n t No. 1-6, p a p e r p r e s e n t e d a t t h e AHS
N a t i o n a l S p e c i a l i s t ' s Meeting on Rotor System Design, P h i l a d e l p h i a , Penn.,
Oct. 22-24, 1980.
8.
Davis, J . M . ; B e n n e t t , R. L.; and Blankenship, B. L.:
R o t o r c r a f t F l i g h t Simulat i o n w i t h A e r o e l a s t i c Rotor and Improved Aerodynamic R e p r e s e n t a t i o n , Vol. 1 E n g i n e e r ' s Manual, USAAMRDL-TR-74-1OA, J u n e 1974.
9.
M i l i t a r y S p e c i f i c a t i o n , F l y i n g Q u a l i t i e s of P i l o t e d A i r p l a n e s , MIL-F-8785B(ASG),
Aug. 1969.
10.
Jewel, J r . , J . W . ; and Heyson, H. H.:
C h a r t s of Induced V e l o c i t i e s Near a L i f t i n g R o t o r , NASA MEMO 4-15-59LY May 1959.
-?
46
2. Govmmrnt Acclrion
1. Report No.
No.
3. Rucipient's Catalog No.
NASA TM-84281
5. Report Date
4. Title and Subtitle
SeDtember 1982
A MATHEMATICAL MODEL OF A SINGLE MAIN ROTOR
HELICOPTER FOR PILOTED SIMULATION
6. Performing Orgniration codr
8. Performing Organization Report No.
7. Author(s1
P e t e r D. T a l b o t , Bruce E. T i n l i n g ,
W i l l i a m A. Decker, and Robert T. N . Chen
b
r
,,
A-9033
10. Work Unit No.
9. PerformingOrganizationName and Address
T-6292Y
NASA Ames Research C e n t e r
M o f f e t t F i e l d , C a l i f o r n i a 94035
I
8
No.
11. Contract or Grant
. 13. Type of Report and Pari4 Cotmod
12. Sponsoring Agency Name and Address
T e c h n i c a l Memorandum
N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n
Washington, D.C. 20546
14. Sponrorir) Agency
Code
15. Supplementary Notes
P o i n t of Contact: W i l l i a m A. Decker, Mail S t o p 211-2, NASA Ames Research
C e n t e r , M o f f e t t F i e l d , CA 94035 (415) 965-5362 o r FTS 448-5362
~~
~~
16. Abstract
T h i s r e p o r t documents a h e l i c o p t e r mathematical model s u i t a b l e f o r
p i l o t e d s i m u l a t i o n of f l y i n g q u a l i t i e s . The mathematical model i s a nonl i n e a r , t o t a l f o r c e and moment model of a s i n g l e main r o t o r h e l i c o p t e r . The
model h a s t e n d e g r e e s of freedom:
s i x rigid-body, t h r e e r o t o r - f l a p p i n g , and
t h e r o t o r r o t a t i o n a l d e g r e e s of freedom. The r o t o r model assumes r i g i d
b l a d e s w i t h r o t o r f o r c e s and moments r a d i a l l y i n t e g r a t e d and summed a b o u t
the azimuth.
The f u s e l a g e aerodynamic model u s e s a d e t a i l e d r e p r e s e n t a t i o n
o v e r a nominal a n g l e of a t t a c k and s i d e s l i p r a n g e of +15", and i t u s e s a
s i m p l i f i e d c u r v e f i t a t l a r g e a n g l e s of a t t a c k o r s i d e s l i p . S t a b i l i z i n g
s u r f a c e aerodynamics a r e modeled w i t h a l i f t c u r v e s l o p e between s t a l l
l i m i t s and a g e n e r a l c u r v e f i t f o r l a r g e a n g l e s of a t t a c k . A g e n e r a l i z e d
s t a b i l i t y and c o n t r o l augmentation system i s d e s c r i b e d . A d d i t i o n a l computer
s u b r o u t i n e s p r o v i d e o p t i o n s f o r a s i m p l i f i e d engine/governor model, atmop h e r i c t u r b u l e n c e , and a l i n e a r i z e d six-degree-of-freedom
dynamic model f o r
s t a b i l i t y and c o n t r o l a n a l y s i s .
7. Key Words 6uggestd by Authorlsl)
Helicopter
F l i g h t s i m u l a t i o n ; Handling q u a l i t i e s
H e l i c o p t e r aerodynamics
H e l i c o p t e r f l i g h t dynamics
H e l i c o p t e r s t a b i l i t y and c o n t r o l augm e n t a t i o n sy st e m
Unclassified
18. Distribution Statement
Unlimited
Unclassified
S u b j e c t Category
I
55
*For =le by the National Technical Information Stvice. Springfield, Virginia 22161
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I
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08
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