zotero-db/storage/ETS4PILF/.zotero-ft-cache

SA Technical Pa
evelopment and Validation iloted Simulation elicopter and
External Sling Load
J. D. Shaughnessy Langley Research Center, Hampton, Virginia Thomas N. Deaux Sperry Sapport Services, Hampton, Virginia Kenneth R. Yenni Langley Research Center, Nampton, Virginia
National Aeronautics and Space Administration
Scientific and Technical Jnformation Office
1979

CONTENTS

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

SYmOLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

D

EAACCEERFELLSCounxotuqooomaRtagstuotnedoedoIioartrP-nlmrdrtsGnaTSeioipaagoSIrunlOhtoleDnysai-ueNSpsscLytnrAtyeenoideOFosnceaamStFmrslfdeM CoyiimiogMMdscooAntyh.AsndeoneTtemttaS..arHliaCnooymsEc.nd.d.MsotiytcnA..Me.Gn..sTtomao.rI..d.Cm..oveA..eli..l.L..crS..sn...M.y..o..sO..r..t...De..M ..E.m...L...o.....d.M ....e....l..o......d....e.......l..............................................................................................................................................................................................................

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12 12 13 16 17 18 25 26 28 28 30 33

. . . . . . . APPLICATION TO U.S. ARMY CH-54 HELICOPTER AND CARGO CONTAINER

35

S

I

MULATION Computer Cockpit Simulati Visual L Load/Lan Trim Cal

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culations

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39 39 40 42 44 44 46

VESM PRiiIamFltIohuCtelA'amsTtIaiOCotiNoncmaAVlmNeeDMrniotVsfdAiecLl I.aDVtA.iaoTl.IniOd.N.a t...i o...n...................................................................................... . .

47 47 47 50

CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

iii

SUMMARY
A generalized, real-time, p i l o t e d , visual simulation of a single-rotor helicopter, suspension system, and external load i s described and validated f o r t h e f u l l f l i g h t envelope o f t h e U.S. Army CH-54 h e l i c o p t e r and cargo c o n t a i n e r as an example. The mathematical model d e s c r i b e d u s e s modified n o n l i n e a r class i c a l rotor theory f o r both t h e main rotor and t a i l rotor, n o n l i n e a r f u s e l a g e aerodynamics, an elastic suspension system, nonlinear load aerodynamics, and a load-ground c o n t a c t model. The implementation of t h e mathematical model on a l a r g e d i g i t a l computing system i s d e s c r i b e d , and v a l i d a t i o n of the s i m u l a t i o n i s d i s c u s s e d . The mathematical model i s v a l i d a t e d by comparing measured f l i g h t data w i t h simulated d a t a , by comparing l i n e a r i z e d system matrices, e i g e n v a l u e s , and eigenvectors w i t h manufacturers' d a t a , and by t h e s u b j e c t i v e comparison of
handling c h a r a c t e r i s t i c s by experienced p i l o t s . A v i s u a l l a n d i n g d i s p l a y syst e m f o r use i n simulation which g e n e r a t e s t h e p i l o t ' s forward-looking r e a l world d i s p l a y i s d i s c u s s e d , and a s p e c i a l head-up, down-looking l o a d / l a n d i n g zone d i s p l a y i s described.
INTRODUCTION
Simulation and a n a l y t i c a l research has been conducted a t the Langley Research Center (LaRC) t o compare various c o n t r o l system concepts f o r improving the handling q u a l i t i e s of single-rotor helicopters carrying relatively large e x t e r n a l s l i n g l o a d s . These c o n c e p t s i n c l u d e c o n t r o l jets a t t h e l o a d , a mova b l e c a b l e attachment p o i n t on t h e h e l i c o p t e r , and c a b l e a n g l e feedback i n t o the helieopter s t a b i l i t y augmentation system. It w a s believed t h a t the m o s t c o s t - e f f e c t i v e and s a f e way t o compare and s t u d y t h e s e systems w a s through t h e use of a p i l o t e d v i s u a l ' s i m u l a t i o n i n which wide v a r i a t i o n s i n parameters and concept o p t i m i z a t i o n could be e x p l o r e d e a s i l y and q u i c k l y .
L i t e r a t u r e s e a r c h e s and d i s c u s s i o n s w i t h i n d u s t r y d i d n o t l o c a t e any simul a t i o n s f o r a h e l i c o p t e r s l i n g l o a d or any mathematical models having f u l l flight-envelope c a p a b i l i t y f o r a helicopter and load. Therefore such a m o d e l and simulation had t o be developed.
The purpose o f t h i s r e p o r t i s t o d e s c r i b e t h e mathematical model of a h e l i copter and e x t e r n a l s l i n g load which w a s developed, to describe the implementat i o n o f t h e model t o o b t a i n a p i l o t e d v i s u a l s i m u l a t i o n , and t o d e s c r i b e the v a l i d a t i o n o f t h e model and s i m u l a t i o n through comparison of simulated d a t a w i t h a c t u a l f l i g h t d a t a by means o f a n a l y t i c a l t e c h n i q u e s and experienced p i l o t s ' evaluations.
The o v e r a l l mathematical model i s made up o f numerous submodels t h a t d e s c r i b e v a r i o u s components of t h e t o t a l dynamic system. The scope of t h e s e submodels i s d e s c r i b e d i n g e n e r a l terms as f o l l o w s :

Atmospheric model - The atmospheric model has v a r i a b l e a i r d e n s i t y ,
winds with v a r i a b l e magnitude and d i r e c t i o n , and v a r i a b l e - i n t e n s i t y turbulence e
C o n t r o l system model - The h e l i c o p t e r c o n t r o l system model c o n v e r t s
p i l o t ' s cyclic-control-stick, collective-stick, and pedal inputs i n t o main- and t a i l - r o t o r c y c l i c and c o l l e c t i v e - p i t c h i n p u t s .
- Rotor models Nonlinear models f o r t h e main and t a i l r o t o r s d e f i n e
t h r u s t , d r a g , and s i d e f o r c e s as w e l l as hub and f o r c e moments r e p r e s e n t a t i v e of a r t i c u l a t e d rotors over a wide range of a i r s p e e d s from 100 knots through hover t o rearward and sideward f l i g h t t o a t l e a s t 20 k n o t s . The rotor models account f o r variable inflow velocity, variable rotor speed, blade t w i s t , t i p loss, b l a d e coning, b l a d e f l a p p i n g , a c t u a t o r dynamics, flapping-hinge o f f s e t , and tail-rotor flapping-hinge cant angle.
- Automatic f l i g h t c o n t r o l system (AE'CS) model The h e l i c o p t e r AFCS
model i s based on t h e system used i n t h e U.S. Army CH-54 h e l i c o p t e r . This AFCS p r o v i d e s h e l i c o p t e r r a t e and a t t i t u d e s t a b i l i z a t i o n i n roll, p i t c h , and yaw.
Fuselage aerodynamics model - The f u s e l a g e aerodynamics model d e f i n e s
nonlinear l i f t , drag, and s i d e f o r c e s as well as p i t c h i n g , r o l l i n g , and yawing moments i n terms of a wide range of f u s e l a g e a n g l e s of a t t a c k and s i d e s l i p , r o t o r downwash, body a n g u l a r r a t e s , and dynamic p r e s s u r e .
E x t e r n a l - l o a d aerodynamics model - A n e x t e r n a l - l o a d aerodynamics
model d e f i n e s n o n l i n e a r l i f t , drag, and s i d e f o r c e as w e l l a s p i t c h i n g , r o l l i n g , and yawing moments as a f u n c t i o n of a n g l e s o f a t t a c k and s i d e s l i p , dynamic p r e s s u r e , r o t o r downwash, and body a n g u l a r r a t e s .
Load suspension model - A l o a d suspension model d e f i n e s c a b l e t e n s i o n
i n one o r more c a b l e s and t h e r e s u l t i n g f o r c e s and moments a c t i n g on t h e h e l i c o p t e r and e x t e r n a l load.
- Load-ground c o n t a c t model A load-ground c o n t a c t model determines
t h e approximate f o r c e s and moments a c t i n g on t h e l o a d as it comes i n contact with the ground f o r pickup and landing.
A general s e t of nonlinear, rigid-body equations of motion €or both t h e helicopter and external load determines the motion of each vehicle with respect t o a f l a t , n o n r o t a t i n g Earth. An algorithm determines t h e trimmed h e l i c o p t e r c o n t r o l p o s i t i o n s , h e l i c o p t e r a t t i t u d e , and load p o s i t i o n and a t t i t u d e so t h a t t h e e n t i r e dynamic system i s i n unaccelerated f l i g h t f o r a s p e c i f i e d i n i t i a l f l i g h t condition. Another algorithm o b t a i n s the equivalent l i n e a r system from t h e n o n l i n e a r model once t h e h e l i c o p t e r i s trimmed; t h e l i n e a r system i s used f o r v e r i f i c a t i o n and validation only.
The mathematical model i s programed on a general-purpose, r e a l - t i m e d i g i t a l computer, w i t h d a t a f o r t h e U.S. Army CH-54 h e l i c o p t e r used as i n p u t s ; and appropriate outputs are fed t o a cockpit having a set of f l i g h t instruments. The computer o u t p u t s a l s o d r i v e a real-world, out-the-window v i s u a l d i s p l a y as

well as a computer-generated l o a d / l a n d i n g zone d i s p l a y f o r p i l o t use. The research p i l o t i s a b l e t o c o n t r o l t h e simulated h e l i c o p t e r by making c y c l i c and c o l l e c t i v e - s t i c k and pedal i n p u t s i n t h e cockpit which generate electric a l s i g n a l s t h a t are transmitted t o t h e computer. Finally, provisions a r e m a d e f o r recording simulated f l i g h t data and f o r i n t e r a c t i n g with the simulation from a c o n t r o l console.
U s e of t r a d e names o r names of manufacturers i n t h i s r e p o r t does n o t cons t i t u t e an o f f i c i a l endorsement of such products o r manufacturers, e i t h e r expressed o r implied, by t h e National Aeronautics and Space Administration.
C o n t r i b u t i o n s t o t h i s work and a s s i s t a n c e t o t h e a u t h o r s w e r e provided by t h e f o l l o w i n g p e r s o n s : Dean E. Cooper, of Sikorsky A i r c r a f t D i v i s i o n of United Technologies Corporation; NASA t e s t p i l o t P e r r y L. Deal; L t . C o l . W. L. Welter, o f Langley Directorate, USAAMRDL; p i l o t s from Evergreen H e l i c o p t e r , Incorpor a t e d , McMinnville, O r e . , Colonial Helicopters, Incorporated, Norfolk, Va., and the 355th Aviation Company a t F o r t E u s t i s , V i r g i n i a ; W. F. L o v e l l , J. B. L e a v i t t , and L. E. Becker, of Sperry Support S e r v i c e s ; M. D. Pardue, g r a d u a t e s t u d e n t a t Old Dominion U n i v e r s i t y ; and Lawrence E. B a r k e r , Jr., Lemuel E. Meetze, and Richard E. Bardusch, of LaRC.

SYMBOLS

Measurements, c a l c u l a t i o n s , and programing w e r e made i n U.S. Customary Units. They are p r e s e n t e d h e r e i n t h e I n t e r n a t i o n a l System of U n i t s ( S I ) .

A~~ B~~

r o t o r 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 c o n t r o l commands, defined by equations (121, rad

AICafcs 1 'ICafcs

r o t o r 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 AFCS commands,
defined by e q u a t i o n s (14) and (13) , r a d o r deg

A;C B;C

r o t o r l a t e r a l and longitudinal c y c l i c c o n t r o l displacements, defined by equations ( 2 3 ) , r a d

a

rotor blade lift-curve slope, per rad

a'

s m a l l angle t h a t d e f i n e s t h e r o t o r d r a g f o r c e , defined by

equation (35), rad

ax, htay,htaz,h

h e l i c o p t e r body-axes a c c e l e r a t i o n s, m/sec2

a0

r o t o r coning angle given by equation ( 2 9 ) , rad

a1'b l

r o t o r longitudinal and lateral flapping angles with respect to c o n t r o l axes, defined by equations ( 3 2 ) , rad

als'bls

r o t o r longitudinal and lateral flapping angles with respect t o s h a f t axes, d e f i n e d by e q u a t i o n s (431, r a d o r deg

B

rotor-blade tip-loss constant

3

b

number of b l a d e s per r o t o r

h' "R

Euler angle transformation matrix f o r helicopter and load,

d e f i n e d by e q u a t i o n (1)

Q '

r o t o r torque c o e f f i c i e n t , d e f i n e d by equation (36)

CT

rotor t h r u s t coefficient, defined by equation (27)

CY

r o t o r side-force c o e f f i c i e n t , defined by equation ( 3 8 )

C

rotor blade chord, m

dlc,eld2c,e rd3c ,e

cable d i r e c t i o n c o s i n e s defined by equations (73)

ekf e k t

fuselage and t a i l angle-of-attack corrections due to rotor downwash, r a d

emr

r o t o r downwash f a c t o r , d e f i n e d by e q u a t i o n (521, rad

e

rotor f lapping-hinge o f f s e t, m

CFxlh' CFy,h r CFz,h'

CFxlRJFy, !LJFZ, R
1 GB(3' GBql GBxrGA@

GAP,GAX,GOt~lGOtr,

GOch

J

f o r c e summations along h e l i c o p t e r and load body axes i n c l u d i n g a l l e x t e r n a l f o r c e s due t o r o t o r , body aerodynamics, ground contact, and suspension system, N
AFCS feedback g a i n s (see t a b l e I f o r units)

G ~ ~ ~ , K ~ ~ ~ ~ e,ngKine~/g~ov~ern, orK p,arameters (see table I f o r u n i t s )

Gu1 Gv1 Gw

g u s t v a r i a b l e s d e f i n e d by e q u a t i o n s (111, m / s e c 2

9

a c c e l e r a t i o n of g r a v i t y , m/sec2

hhrh!L 'bmr 'bt

helicopter and load a l t i t u d e , m main- and t a i l - r o t o r b l a d e f l a p p i n g moment of i n e r t i a , kg-m 2

Imr

main-rotor p o l a r moment of i n e r t i a , kg-mL

IPt

engine-power-turbine moment of i n e r t i a , kg-m2

Ixx, h1Iyy, h r l z z ,h'
I x x ,R I Iyy ,R 1 = z z ,R

h e l i c o p t e r and l o a d m a s s moments o f i n e r t i a about
body a x e s , kg-m2

4

'xz,h, Ixz,R

h e l i c o p t e r and l o a d p r o d u c t s o f i n e r t i a , kg-m 2

it

flow i n c i d e n c e a t h e l i c o p t e r h o r i z o n t a l t a i l , d e f i n e d by

equation (541, r a d or deg

=to

f i x e d incidence of h e l i c o p t e r h o r i z o n t a l t a i l , rad

1 Kco Kcl Kc2 'Kc 3
Kc4 Kc5 Kc6 %7

control-system s t i c k gain, control mixing, and u n i t conversion constants (see table I f o r u n i t s )

Kfe

f u s e l a g e pitching-moment parameter due t o rotor t h r u s t used i n

e q u a t i o n s (581, N-m

%

load-ground contact position force parameter used i n

e q u a t i o n (61), N/m

Ksc

l o a d suspension system cable s p r i n g r a t e , N/m

Kv

load-ground c o n t a c t v e l o c i t y force parameter used i n equation (611,

N/ ( m / s e c )

Lc,RtMc,Q' Nc,R

t o t a l load-ground c o n t a c t moments, d e f i n e d by e q u a t i o n s ( 6 6 ) , N-m

Lei, R' M c i ,R rN c i ,R Ld,h' Md, h 'Nd,h

l o a d moments due t o ground c o n t a c t a t p o i n t i , N-m
h e l i c o p t e r body-axes moments due t o body a n g u l a r rates, d e f i n e d by e q u a t i o n s (891, N-m

Ld, Rr'd, krNd,k

l o a d moments due to l o a d a n g u l a r v e l o c i t i e s , d e f i n e d by e q u a t i o n s (951, N-m

Lf,h'Mf,hlNf,h

t o t a l moments a c t i n g on h e l i c o p t e r due t o fuselage aerodynamics, d e f i n e d by e q u a t i o n s ( 5 8 ) , N-m

rijh I i;h Lhub, htMhub,h' Nhub,h

h e l i c o p t e r fuselage l i f t , drag, and s i d e forces, N
h e l i c o p t e r moments i n body axes due t o r o t o r moments t r a n s m i t t e d a t t h e hub, d e f i n e d by e q u a t i o n ( 4 5 ) , N-m

Lhub,s,Mhub,s,Nhub,s

h e l i c o p t e r moments i n s h a f t axes due t o r o t o r moments t r a n s m i t t e d a t t h e hub, defined by equations (42) and (44), N-m

t o t a l aerodynamic moments a c t i n g on t h e l o a d , d e f i n e d by e q u a t i o n s ( 9 6 ) , N-m

l i f t , drag, and s i d e f o r c e s a c t i n g on t h e load, defined by equations (921, N

5

Lr,h P M,rh‘ Nr,h

t o t a l body-axes moments a c t i n g on h e l i c o p t e r due t o
r o t o r f o r c e s and hub moments, d e f i n e d by e q u a t i o n (46) ,
N-m

Lt,h r M t ,h r N t ,h

body-axes moments a c t i n g on h e l i c o p t e r ‘due t o suspension c a b l e , d e f i n e d by e q u a t i o n s (751, N-m

Lt,!PM,tR, N t ,R

body-axes moments a c t i n g on t h e l o a d due t o suspension
cable, d e f i n e d by e q u a t i o n s (77) , N-m scales o f turbulence, d e f i n e d by equations ( 5 ) t o (7) , m

body-axes aerodynamic moments a c t i n g on f u s e l a g e , appearing i n e q u a t i o n s (581, N-m

e x t e r n a l moments, i n c l u d i n g main- and t a i l - r o t o r moments,
f u s e l a g e aerodynamic moments, and c a b l e suspension
moments, N-m

RC

suspension-system i n s t a n t a n e o u s c a b l e l e n g t h , d e f i n e d by

equation (69), m

Rc 0

suspension-system unstretched cable length, m

%?

m a s s moment of r o t o r blade a p p e a r i n g i n e q u a t i o n s (42), kg-m

mhrmR

h e l i c o p t e r and l o a d m a s s , kg

- n11n2,. * 1ng

white-noise signals used i n equations (10)

phl q h l ‘h 1PR ‘2 19%1

h e l i c o p t e r and l o a d a n g u l a r v e l o c i t y components i n body axes, rad/sec

Psr qs 1rs

h e l i c o p t e r angular v e l o c i t y components i n s h a f t
axes, rad/sec

Pwrqwl rw

h e l i c o p t e r angular v e l o c i t y components i n c o n t r o l
axes , rad/ s e c

Qam

aerodynamic t o r q u e a c t i n g on main r o t o r , p o s i t i v e i n d i r e c t i o n

o p p o s i t e t o r o t a t i o n , N-m

Qat

aerodynamic torque a c t i n g on t a i l rotor, p o s i t i v e i n d i r e c t i o n
o p p o s i t e t o r o t a t i o n , N-m

Qeng

s h a f t engine t o r q u e a c t i n g on r o t o r and f u s e l a g e ( p o s i t i v e v a l u e t e n d s t o accelerate r o t o r and cause f u s e l a g e t o yaw r i g h t ) , N-m

Qgen QS

gas-generator t o r q u e used i n e q u a t i o n s (481, N-m r e s u l t i n g main- ox t a i l - r o t o r s h a f t torque a c t i n g on f u s e l a g e , N-m

6

--
qhf qR

dynamic pressure a t f u s e l a g e and l o a d , given by e q u a t i o n s (50)
and (91) , N/m2

R

main- o r t a i l - r o t o r r a d i u s , m

S

s h a f t tilt transformation mafrix given by equation (18)

S

Laplace o p e r a t o r I sec-1

T,H, J

main- o r t a i l - r o t o r t h r u s t , d r a g , and s i d e f o r c e d e f i n e d by
, e q u a t i o n s ( 2 8 ) , ( 3 4 ) and ( 3 9 ) , N

%

suspension-system c a b l e tension, defined by equation (70) , N

Uas,h Vas,h t w a s ,h

h e l i c o p t e r body-axes components of v e l o c i t y , defined by
e q u a t i o n s (20) , m/sec

Uas,RfVas,QfWas,Q

load body-axes components of t o t a l a i r s p e e d , defined by
e q u a t i o n s (90) , m/sec

Ucg ,h vcg ,h wcg,h

body-axes components of i n e r t i a l v e l o c i t y of h e l i c o p t e r c.g., given by e q u a t i o n s ( 8 0 ) , m/sec

ucg ,R vcg ,R wcg,R

f

f

body-axes components o f i n e r t i a l v e l o c i t y of load
c.g., given by e q u a t i o n s (80) w i t h h replaced
by R, m / s e c

uc i ,e' vci, e w c i,e

Earth-axes components of i n e r t i a l v e l o c i t y of t h e i t h load-ground c o n t a c t p o i n t , defined by equation (601, m/sec

Uhcg, e' Vhcg,e ' whcg,e

Earth-axes components of i n e r t i a l v e l o c i t y of h e l i c o p t e r c.g., given by e q u a t i o n (841, m/sec

UQcgI e tVRcg WRcg,e

Earth-axes components of i n e r t i a l v e l o c i t y of load c.g., given by e q u a t i o n (84) w i t h h r e p l a c e d by 2, m/sec

Ugust,R V g u s t,R' W g u s,t R f

body-axes components of g u s t
velocity, given by equations (90) , m / s e c

ur,h vr,h 'wr,h

body-axes components o f t o t a l a i r s p e e d a t r o t o r hub, given by equations (191, m / s e c

shaft-axes components of t o t a l a i r s p e e d a t rotor hub, given by e q u a t i o n (171, m/sec

uwfvwf ww

control-axes components of t o t a l a i r s p e e d a t rotor hub, given by equations (221, m / s e c

%ind ,e Vwind,e Wwind,e

Earth-axes components of windspeed, given by equation (4) I m/sec
7

Uwind ,h Vwind,h' Wwind,h' Uwind ,R' Vwind,R' Wwind,

body-axes components of windspeed for h e l i c o p t e r
and l o a d , given by e q u a t i o n ( 4 ) , m/sec

'as, hl'as,

h e l i c o p t e r and load t o t a l airspeed defined by equat i o n s (51) and (911, m / s e c

'wind
VO
h
was,R

magnitude of wind v e c t o r , m / s e c
l o a d s l i d i n g v e l o c i t y beyond which s l i d i n g f r i c t i o n becomes independent of s l i d i n g v e l o c i t y , m/sec
l o a d body z-axis component of t o t a l a i r s p e e d w i t h c o n s t a n t added t o improve numerical s t a b i l i t y , m/sec

'ci ,e "ci ,e 1 'ci ,e

Earth-axes components of load-ground contact f o r c e s a t the i t h contact p o i n t , defined by equations (611,
(62) , and (631, N

Xci,R,Yci,R,Zci,R

load body-axes components of load-ground c o n t a c t f o r c e s a t t h e i t h c o n t a c t p o i n t , given by equation (641, N

Xc,RJCIR, Zc,R

load body-axes components of t o t a l load-ground contact f o r c e s , given by equations (661, N

Xf,hlYf,h'2f , h

h e l i c o p t e r body-axes components of fuselage l i f t , drag, and s i d e f o r c e , given by e q u a t i o n (571, N

XR' YR' 2%
r' ,h1'r1 h' zr ,h

l i f t , drag, and s i d e f o r c e s expressed i n load body axes (eq. ( 9 3 ) ) N
h e l i c o p t e r body-axes components o f r o t o r f o r c e s , given
by equation (411, N

'r, s,Yr,s,zrs,

shaft-axes components of r o t o r f o r c e s , defined by equations (401, N

't ,h t Y t,h r't ,h

h e l i c o p t e r body-axes components of cable t e n s i o n , given
by equation (74) , N

X,tR J t ,R J t , R

l o a d body-axes components of c a b l e t e n s i o n , given by
equation (76) , N

xa ,h 1ya ,h1za ,h

h e l i c o p t e r body-axes d i s t a n c e s from h e l i c o p t e r c.g. t o cable attachment point, m

xalR'Ya,R'Za, R

load body-axes d i s t a n c e s from load c.g. t o cable a t t a c h ment p o i n t , m

Xc i ,elYci ,e' z c i , e

Earth-axes coordinates of i t h ground contact p o i n t , m

8

Xc i , R' Y c i , R' 'ci ,R
Xc o l ' x l a t ' x l o n Xped
Xha ,e Yha,e' 'ha, e
Xhcg r e"hcg,e' Zhcg,e

Xkeg, e 'Ytcg, e ZRcg,e

Xkcg ,h' 'Reg ,h' 'Reg,h

XR p ,h'

h' 'Rp ,h

XLp ,R 'Y R p ,R

R

Xmr ,htYmr,h"mr, h'
X t rh, r Y t r, h t Z t r,h
Xp s , h r Y p s ,h"ps, h

Xscreen1 Yscreen

xv, e 1 Yv, e 1 zv, e
xv, h yv ,h zv, h Xw t , h l 'wt, h' 'wt ,h Xzp ,e, Yzp ,e ,'zp ,e

load body-axes d i s t a n c e s f r o m load c,g. t o i t h ground contact point, m
control displacements of collective s t i c k , l a t e r a l and longitudinal c y c l i c s t i c k , and pedals (positive displacements cause climb, r i g h t r o l l , p i t c h - u p , and yaw left, respectively), m
Earth-axes i n e r t i a l p o s i t i o n of h e l i c o p t e r c a b l e a t t a c h ment p o i n t , given by equation ( 6 7 ) , m
Earth-axes i n e r t i a l position of helicopter c.g., given by e q u a t i o n s (851, m
Earth-axes i n e r t i a l p o s i t i o n of load cable attachment
p o i n t , given by equation (68) , m
Earth-axes i n e r t i a l p o s i t i o n o f load c.g., given by
equations (85) with s u b s c r i p t h replaced by 2 , m
h e l i c o p t e r body-axes coordinates o f load c.g., given by equation (971, m
h e l i c o p t e r body-axes c o o r d i n a t e s of any p o i n t p on t h e load, given by equation (98), m
l o a d body-axes c o o r d i n a t e s of any p o i n t p on t h e load, m
h e l i c o p t e r body-axes c o o r d i n a t e s of main- and t a i l - r o t o r hub, m
h e l i c o p t e r body-axes coordinates of eye-level p o s i t i o n of p i l o t i n l e f t seat, m
nondimensional screen coordinates of any p o i n t p , used i n load/landing zone display defined by equat i o n s (1011, m
Earth-axes coordinates of helicopter viewpoint, given by e q u a t i o n (991, m
h e l i c o p t e r body-axes coordinates of down-looking viewpoint, m
h e l i c o p t e r body-axes coordinates o f wind-tunnel mounting point, m
Earth-axes i n e r t i a l p o s i t i o n of any p o i n t p f i x e d on the Earth, m

9

, , Xzp h 1 Yzp h 1 'zp ,h

h e l i c o p t e r body-axes coordinates of any p o i n t p on the Earth, m

ac ,e ,'c, e

cable angles with respect t o t h e v e r t i c a l , defined by equations (71)
and (72), rad

af 1 %
af R
B!L
P Y

fuselage angle of a t t a c k and s i d e s l i p , defined by equations (49) and (55), rad or deg
local fuselage angle of a t t a c k , defined by equation (53) , r a d or deg
l o a d a n g l e s of a t t a c k and s i d e s l i p , d e f i n e d by e q u a t i o n s (91), r a d

rotor orientation angle, rad

r o t o r Lock number,

pacR4 Rotor b l a d e f l a p p i n g moment of i n e r t i a

azlh1az2h
n'lwt n'2wt At

incremental fuselage l i f t , N i n c r e m e n t a l f u s e l a g e p i t c h i n g moment, N-m
simulation integration step size, sec

63,

tail-rotor flapping-hinge cant angle, rad

0ct

commanded v a l u e o f t a i l - r o t o r c o l l e c t i v e p i t c h given by
equations (12) , rad

@Om 00,

main-rotor c o l l e c t i v e p i t c h , given by e q u a t i o n s ( 1 2 ) , r a d o r deg
e f f e c t i v e value o f t a i l - r o t o r c o l l e c t i v e p i t c h , given by equation (471, rad o r deg

'mafcs, ' t a f c s

main- and t a i l - r o t o r AFCS c o l l e c t i v e p i t c h command, d e f i n e d by e q u a t i o n s (15) and (16), r a d

@S

l o n g i t u d i n a l s h a f t tilt angle, p o s i t i v e f o r t h r u s t vector t i l t e d

from -zh toward -xh, r a d

01 0 . 75

rotor blade t w i s t angle from r o o t t o t i p , r a d rotor c o l l e c t i v e p i t c h a t three-fourths radius, defined by
equation (33), rad

'h,$hf$h

h e l i c o p t e r p i t c h , r o l l , and yaw AFCS a t t i t u d e e r r o r s , r a d

Am, A t
Vf

main- and tail-rotor inflow r a t i o , defined by equation (25) c o e f f i c i e n t of s l i d i n g f r i c t i o n used i n load-ground c o n t a c t model

10

main- and t a i l - r o t o r t i p - s p e e d r a t i o , d e f i n e d by e q u a t i o n (24)
r o t o r induced inflow r a t i o , defined by equation (26)
atmospheric d e n s i t y g i v e n by e q u a t i o n ( 2 ) , kg/m 3

swashplate a c t u a t o r damping r a t i o

rotor s o l i d i t y , b-c 7TR

body-axes components of r m s g u s t i n t e n s i t y , m / s e c

engine t i m e c o n s t a n t , sec

t a i l - r o t o r ti,, t i m e c o n s t a n t , sec

r o t o r inflow t i m e c o n s t a n t , sec

white n o i s e power s p e c t r a l d e n s i t y , sec

@hlehl$h,$g,efi,$g

r o l l ($), p i t c h ( e ) , and yaw ($1 E u l e r a n g l e s o f
helicopter and load, rad

@S

lateral s h a f t tilt angle, p o s i t i v e f o r t h r u s t vector t i l t e d from

-zh toward +yh, r a d

@wind

wind a n g l e measured clockwise from t r u e n o r t h t h a t d e f i n e s t h e d i r e c t i o n from which t h e wind i s coming, r a d

@wt
R

wind-tunnel yaw a n g l e , r a d rotor angular velocity, rad/sec

QO

commanded r o t o r a n g u l a r v e l o c i t y , r a d / s e c

power-turbine angular v e l o c i t y , rad/sec QPt

wa

swashplate-actuator natural frequency, rad/sec

Abbreviations :

ADC

analog to d i g i t a l converter

AFCS

automatic f l i g h t control system

c.g.

center of gravity

DAC

d i g i t a l to analog converter

11

BLCG

buttock l i n e a t c,g.

FSCG

fuselage s t a t i o n a t c.g.

WLCG

w a t e r l i n e a t c.g.

RTS

real-time system

IXIS

r o o t mean squared

VLDS

visual landing display system

Dots o v e r a symbol denote d e r i v a t i v e s w i t h r e s p e c t t o t i m e .

I n i t i a l v a l u e s of v a r i a b l e s are denoted by (0) f o l l o w i n g t h e v a r i a b l e symbol.

Matrix t r a n s p o s e i s denoted by a s u p e r s c r i p t T.

DESCRIPTION O F MATHEMATICAL MODEL
The mathematical model f o r t h e s i m u l a t i o n of a s i n g l e - r o t o r h e l i c o p t e r and e x t e r n a l l o a d can be given i n terms o f submodels f o r t h e v a r i o u s components of t h e t o t a l dynamic system.
A block diagram f o r t h e o v e r a l l mathematical model i s given i n f i g u r e 1. This diagram together with the Contents should be useful i n understanding the i n t e r r e l a t i o n s h i p between t h e submodels. I n t h e following s e c t i o n s t h e mathematical model i s d e s c r i b e d i n approximately t h e o r d e r it would be executed i n t h e computer so t h a t t h e r e a d e r can w r i t e h i s own s i m u l a t i o n program more easily.

Coordinate Systems
The f o l l o w i n g right-hand o r t h o g o n a l a x i s systems are used i n t h i s r e p o r t :
(1) E a r t h a x e s ( s u b s c r i p t e ) : o r i g i n f i x e d on t h e E a r t h ' s s u r f a c e , xe-axis pointing north, ye-axis pointing east
( 2 ) H e l i c o p t e r body axes ( s u b s c r i p t h) : o r i g i n a t t h e h e l i c o p t e r c.g., xh-axis p o i n t i n g forward i n t h e p l a n e o f symmetry of t h e f u s e l a g e and p a r a l l e l t o t h e h e l i c o p t e r w a t e r l i n e , zh-axis p o i n t i n g downward away from t h e main r o t o r and i n t h e p l a n e o f symmetry (see f i g s . 2 and 3)
( 3 ) Shaft axes ( s u b s c r i p t s ) : o r i g i n a t the c e n t e r of the r o t o r hub, y,-axis r o t a t e d through the lateral s h a f t tilt angle $s about the xh-axis, xs-axis r o t a t e d through the longitudinal s h a f t tilt a n g l e 8, about t h e ys-axis, zs-axis c o i n c i d e n t w i t h t h e r o t o r s h a f t , a p p l i e s t o both t h e main r o t o r and t h e t a i l r o t o r (see f i g . 3)
12

(4) Control axes (subscript w ) : origin a t the center of the rotor hub,
- zw-axis d i r e c t e d toward t h e f u s e l a g e along t h e a x i s of no f e a t h e r -
ing (the physical axis of a pure flapping rotor a rotor with blades f i x e d i n p i t c h b u t free t o f l a p ) , %-axis chosen t o give no yw-component of v e l o c i t y r e l a t i v e t o t h e free-stream a i r ( t h e freestream a i r t o i n c l u d e t h e a i r motion from t u r b u l e n c e and s t e a d y winds), applies t o both r o t o r s (see f i g , 3)
(5) Wind-tunnel axes (subscript w t ) : o r i g i n a t t h e helicopter o r external-load wind-tunnel mounting point, s t - a x i s pointing i n t o t h e r e l a t i v e wind, zwt-axis p o i n t i n g downward and p e r p e n d i c u l a r t o the wind
(6) External-load body axes ( s u b s c r i p t R) : o r i g i n a t t h e load c.g.,
xR-axis p o i n t i n g forward i n t h e p l a n e of symmetry of t h e load, zR-axis p o i n t i n g downward i n t h e p l a n e of symmetry o f t h e l o a d
The Earth-to-body a x i s t r a n s f o r m a t i o n f o r t h e h e l i c o p t e r based on t h e s t a n d a r d yaw, p i t c h , and r o l l E u l e r a n g l e r o t a t i o n sequence shown i n f i g u r e 3 i s given i n m a t r i x form as

s i n 6, s i n $h cos $h
- cos $h s i n $h
s i n oh cos $h cos $h
L + s i n $h s i n $h

s i n 8, s i n $h s i n $h
+ cos @h cos $h s i n 6, cos $h s i n qh
- s i n @h cos $h

-sin 8h cos 8h s i n $h
cos 8, cos $h

The t r a n s f o r m a t i o n m a t r i x f o r t h e l o a d h a s t h e same form w i t h t h e s u b s c r i p t h replaced by s u b s c r i p t 2.

Atmospheric Mode1
The atmospheric model used allows f o r a i r - d e n s i t y v a r i a t i o n s with a l t i tude, v a r i a b l e wind d i r e c t i o n and magnitude, and v a r i a b l e - i n t e n s i t y atmospheric turbulence. The a i r d e n s i t y is c a l c u l a t e d according t o t h e polynomial i n h

- p = 1.2266 (1.176 X 10'4)h 4- (4.337 X 10-')h2 - (7.463 X 10'14)h3

- + (5.538 x 1 0 - - ~ ~ ) h * (9.357 x 1 0 - ~ ~ ) h ~

(2)

where h i s t h e h e l i c o p t e r o r external-load a l t i t u d e above sea l e v e l , i n meters, as determined from i n t e g r a t i n g t h e equations of motions discussed below.
13

Steady winds are s p e c i f i e d i n t e r m s of magnitude and d i r e c t i o n . The wind is expressed i n Earth axes by the r e l a t i o n s

%ind,e -- 'wind 'Os *wind

Vwind,e -- 'wind sin *wind

(3)

Wwind,e =

where *wind i s t h e a n g l e measured clockwise from t r u e n o r t h t h a t d e f i n e s t h e d i r e c t i o n from which t h e wind i s blowing. Steady v e r t i c a l winds are n o t simul a t e d . The components o f t h e s t e a d y winds a r e expressed i n h e l i c o p t e r body axes by using t h e t r a n s f o r m a t i o n

~ n d ,~~n d , ~

Vwind,h = ch Vwind,e

(4)

Wwind,h

Wwind,e

The components o f t h e s t e a d y winds are e x p r e s s e d i n l o a d body axes by u s i n g
e q u a t i o n (4) w i t h t h e s u b s c r i p t h r e p l a c e d by s u b s c r i p t R. The atmospheric t u r b u l e n c e mathematical model i s based on t h e Dryden spectrum o f t u r b u l e n c e as d i s c u s s e d i n r e f e r e n c e 1. This theory a l l o w s t h e s i m u l a t i o n of atmospheric turbulence by passing uncorrelated white noise through l i n e a r f i l t e r s to obtain g u s t components i n body axes.
The scales of t u r b u l e n c e f o r t h e h e l i c o p t e r and t h e l o a d are c a l c u l a t e d according t o the r e l a t i o n s shown f o r t h e h e l i c o p t e r

Lv,h = %,h
(762.0 14

(hh =< 762.0; Ou < 6.4)

(CTU => 6.4)

(7)

(hh > 762.0; Ou < 6.4) J

where 0, is t h e l o n g i t u d i n a l r m s g u s t i n t e n s i t y (assumed t o be t h e same f o r t h e h e l i c o p t e r and l o a d ) , The scales of t u r b u l e n c e f o r t h e load are c a l c u l a t e d by u s i n g e q u a t i o n s ( 5 ) (6) I and ( 7 ) with t h e s u b s c r i p t h r e p l a c e d by subscript A. The v e r t i c a l and l a t e r a l g u s t i n t e n s i t i e s f o r t h e h e l i c o p t e r a r e given by
0.03760,hh 1/3 and
av = au
The g u s t i n t e n s i t i e s f o r t h e l o a d are c a l c u l a t e d from e q u a t i o n s (8) with t h e
s u b s c r i p t h r e p l a c e d by subscript R. The white n o i s e power s p e c t r a l d e n s i t y
is given by

where A t i s t h e d i g i t a l simulation computation s t e p s i z e i n seconds.
The components of g u s t v e l o c i t y f o r t h e h e l i c o p t e r and l o a d i n body
. . axes are c a l c u l a t e d by p a s s i n g s i x u n c o r r e l a t e d white n o i s e s i g n a l s
nlr n2, , I n6 through l i n e a r f i l t e r s and a r e given a s follows f o r t h e helicopter:

Ugust,h --

s

Gulh
+ -'as,

h

nl

) .vIh(. + 'as,h

Vgust l h --

(. fi =v,h n2 +-)

"3

15

where

- m.1 Gu,h - 'u 'as,h (TQoLu,h
Gv,h = v' J3vas,h/(2'QoLv,h)

Gw,h = awJ3vas,h /(2'@o%?,h)

The g u s t components f o r t h e l o a d are computed i n a s i m i l a r manner. The f i l t e r s may be implemented d i g i t a l l y by using Z transform techniques.
I t was judged a f t e r d i s c u s s i o n with G. H. F i c h t l , of t h e NASA Marshall Space F l i g h t Center, t h a t f o r s l i n g - l o a d c a b l e l e n g t h s of approximately 30 meters, t h e g u s t s f o r t h e h e l i c o p t e r and t h e e x t e r n a l load w i l l be essent i a l l y u n c o r r e l a t e d . Thus two sets of u n c o r r e l a t e d white n o i s e s i g n a l s a r e used f o r the helicopter and t h e external load.

Control System

The f l i g h t c o n t r o l system mathematical model c o n v e r t s t h e p i l o t ' s c y c l i c -

s t i c k , pedal, and c o l l e c t i v e - s t i c k displacements and AFCS outputs t o equivalent
main-rotor and t a i l - r o t o r c o n t r o l i n p u t s . The main-rotor c o l l e c t i v e p i t c h eo,

longitudinal and l a t e r a l cyclic p i t c h BIC and ATC, and t a i l - r o t o r c o l l e c t i v e

eCt p i t c h

a r e given i n terms of c o l l e c t i v e , c y c l i c , and pedal displacements

and AFCS o u t p u t s by t h e following e x p r e s s i o n s o b t a i n e d from Sikorsky A i r c r a f t

Division of United Technologies Corporation:

K ~ O+ Kclxcol + 'mafcs
Kc2x10n BICafcs +
Kc3Xcol Kc4xlat + AICafcs +
Kc5 + Kc6xped + Kc7Xcol+ ' t a f c s
where xcol, xlon, xlat, and "ped are t h e p i l o t ' s c o n t r o l displacements w i t h r e s p e c t t o a f i x e d reference p o s i t i o n . The c o n s t a n t s KcO through Kc7 a r e used f o r s t i c k gain, control mixing, and u n i t conversions. Since control f e e l i s important t o p i l o t s , a programmable hydraulic c o n t r o l loader i s used f o r the cyclic control s t i c k and the pedals. Pertinent d e t a i l s of t h i s system are discussed i n the section "Simulation Description."

16

Automatic F l i g h t C o n t r o l System Model
The mathematical mode.1 of t h e automatic f l i g h t c o n t r o l system (AFCS) i s s i m i l a r t o t h a t of t h e a t t i t u d e command system employed i n t h e U.S. Army CH-54A h e l i c o p t e r , and w a s o b t a i n e d from unpublished S i k o r s k y A i r c r a f t d a t a . The p i t c h channel equation is given as

where and

A

h

BICafcs = GBeeh + GBqqh -k GB?lon

ehh = e, - eh(o)

(13)

The q u a n t i t i e s 8,,(0) and xlon(0) are t h e p r e c a l c u l a t e d i n i t i a l t r i m v a l u e s

eh of

and xlon d i s c u s s e d i n t h e s e c t i o n " T r i m C a l c u l a t i o n s . "

The r o l l e q u a t i o n is g i v e n s i m i l a r l y as

and
- h
Xl a t = xl a t xl a t ( 0 )

The yaw channel i s modeled by the f o l l o w i n g e q u a t i o n :

where

A
' t a f c s = G8t$*h + GOtrrh

17

The heading command $J0 i s i n i t i a l l y s e t t o t h e h e l i c o p t e r t r i m yaw angle.

During t h e simulation whenever t h e p i l o t a c t u a t e s e i t h e r microswitch l o c a t e d on

t h e s i m u l a t o r p e d a l s , t h e n $o i s s e t t o t h e c u r r e n t v a l u e of +ha When t h e
p i l o t releases t h e microswitches, q0 i s s e t t o t h e v a l u e of $h j u s t p r i o r

t o switch release. I f the heading hold f e a t u r e i s not desired, then a push

e b u t t o n d i s c r e t e i n t h e c o c k p i t w i l l s e t t h e g a i n G

t o zero.

t$

The a l t i t u d e channel i s modeled by u s i n g t h e f o l l o w i n g e q u a t i o n :

where

The a l t i t u d e command ho i s i n i t i a l l y s e t t o t h e h e l i c o p t e r a l t i t u d e . During t h e o p e r a t e mode, t h e p i l o t can r e s e t ho t o t h e c u r r e n t a l t i t u d e o r disable the a l t i t u d e hold f e a t u r e by using a push-button d i s c r e t e i n t h e cockpit,
R o t o r System
I n t h e i n t e r e s t of real-time pilot-in-the-loop computer simulation, every e f f o r t i s made t o keep t h e r o t o r mathematical model s i m p l e b u t adequate t o allow forward f l i g h t t o a t least 100 knots, rearward and sideward f l i g h t to 2 0 k n o t s , a u t o r o t a t i o n s , and l a r g e - a n g l e maneuvers. The f o r c e s and moments due t o t h e main and t a i l r o t o r s are c a l c u l a t e d by u s i n g a modified B a i l e y represent a t i o n d i s c u s s e d i n r e f e r e n c e s 2 t o 5.
The f o l l o w i n g d i s c u s s i o n of r o t o r modeling i s g e n e r a l and i s a p p l i c a b l e t o b o t h t h e main and t a i l rotors. Important l i m i t a t i o n s are given when necessary. Rotor v a r i a b l e s w i l l need "main ( m ) " o r " t a i l ( t ) "i d e n t i f i c a t i o n when i n c l u d e d i n a computer program.
The v e l o c i t y o f t h e f r e e stream a t t h e hub i s e x p r e s s e d i n s h a f t axes i n t e r m s o f i n e r t i a l v e l o c i t y o f t h e h e l i c o p t e r c e n t e r of g r a v i t y , body angular rates, s t e a d y winds, atmospheric g u s t s , and p o s i t i o n of t h e main- and t a i l r o t o r hubs by t h e following expression:
U S
(17)
vS
wS
18

where

cos e s sin Bs sin Qs sin 0, cos @s

s=

--sin 8,

cos 0, sin @s cos 0, cos Q S

and

- -
Ur,h - %s,h i- qhzr,h rhYr,h

- V r,h

=

v as,h

+

rhxr,h

‘hzr,h

Wr,h - Was,h

phyrIh - qhxr,h

+

and

Uas,h = ucg,h -I-Ugust,h + Uwind,h VasIh = Vcg,h + Vgust,h + vwind,h Waslh = wcg,h + - Wgust,h + Wwind,h

(19)
(20)

The rotor orientation angle B is defined by the relation
- B = tan-1 vs
US
and the velocity of the free stream at the hub is expressed in control axes by the approximation given in reference 2 (see fig. 3(c)) :

% = us cos B + vs sin B

vw = 0

(22)

- - wW = wS BiCUs AiCVs

19

B;. where

and A;C are t h e s h a f t r e l a t i v e c y c l i c c o n t r o l i n p u t s f o r t h e main

A;. r o t o r . For t h e t a i l r o t o r ,

and BiC are o f course zero.

Actuator dynamics are modeled by passing t h e c o n t r o l i n p u t s through secondorder f i l t e r s as

A;C

1 -

wa2

- s2 + 2pawas 4- wa 2 A I C

The f i l t e r s may be implemented by u s i n g Z t r a n s f o r m techniques. The c y c l i c c o n t r o l i n p u t s AIC and BIC are generated by p i l o t c y c l i c s t i c k motions and automatic c o n t r o l system o u t p u t given by equations ( 1 2 ) t o ( 1 4 ) .
The r o t o r f o r c e s and moments are f u n c t i o n s of t h e r o t o r t i p - s p e e d r a t i o and induced i n f l o w r a t i o , as d i s c u s s e d i n r e f e r e n c e s 2 , 4 , 6 , and 7. The t i p speed r a t i o JJ i s given by

J J = - uW
RR
where fi i s t h e r o t o r a n g u l a r v e l o c i t y , The i n f l o w r a t i o
the implicit equation

is calculated by

where t h e induced inflow r a t i o V i s determined from t h e d i f f e r e n t i a l equation

This method o f computing t h e inflow r a t i o

assumes t h a t t h e i n f l o w i s con-

s t a n t a c r o s s t h e r o t o r d i s k . The t h r u s t c o e f f i c i e n t CT i s c a l c u l a t e d below,

and a v a l u e of t h e t i m e c o n s t a n t T A o f approximately 0.1 sec i s chosen t o approximate t h e t i m e l a g associated with change i n rotor inflow. This tech-

nique of using a first-order d i f f e r e n t i a l equation t o calculate the induced

velocity ratio V i s superior t o algebraic calculation, because algebraic

methods were found t o be numerically u n s t a b l e i n d i g i t a l s i m u l a t i o n s ,

20

The r o t o r t h r u s t T and coning angle a. are c a l c u l a t e d t o t h e t h i r d power of t h e tip-speed r a t i o a c c o r d i n g t o t h e f o l l o w i n g r e l a t i o n s taken from r e f e r e n c e 4, where h i g h e r o r d e r terms have been n e g l e c t e d :

where 80 i s t h e e f f e c t i v e blade p i t c h angle a t t h e blade root and 81 i s t h e t w i s t of t h e blade. Then

and

The t e r m i n e q u a t i o n ( 2 2 ) i n r e f e r e n c e 4 i n v o l v i n g t h e b l a d e m a s s moment cont r i b u t e s less t h a n 0.5O, i s e s s e n t i a l l y c o n s t a n t , and is n e g l e c t e d here.
The f u s e l a g e a n g u l a r v e l o c i t y expressed i n c o n t r o l axes i s r e q u i r e d i n t h e c a l c u l a t i o n of c e r t a i n r o t o r f o r c e s and moments. T h i s q u a n t i t y i s o b t a i n e d by rotating the fuselage angular velocity expressed i n s h a f t axes through the
r o t o x o r i e n t a t i o n a n g l e 6, n e g l e c t i n g t h e small c y c l i c p i t c h a n g l e s AIC
and BIC. (See f i g . 3 ( c ) f o r c l a r i f i c a t i o n . )

1 pw = ps COS B + qs s i n B
qw -- -ps s i n B + q, COS 6

(30)

rw = rs

where

21

The f l a p p i n g a n g l e s al and bl are c a l c u l a t e d r e l a t i v e t o c o n t r o l axes by u s i n g t h e f o l l o w i n g formulas d e r i v e d f r o m r e f e r e n c e s 3 , 6, and 8 and unpublished data obtained from Sikorsky Aircraft Division of United Technologies Corporation:

al =

1

1 -- P2

2BZ

and

i

(32)

For a r o t o r b l a d e w i t h l i n e a r t w i s t and c o n s t a n t chord i t can be shown t h a t
replacing eo, t h e blade p i t c h a t the r o o t (appearing i n t h e r e f e r e n c e s ) , with

the p i t c h a t three-fourths radius, and dropping 8 1 w i l l have a negli-

g i b l e e f f e c t on t h e o v e r a l l s o l u t i o n , An e x p r e s s i o n f o r

i s given as

The downwind h o r i z o n t a l component o f t h e r o t o r force i n c o n t r o l axes i s expressed as

H = Ta'

(34)

where t h e small angle a ' i s a function of t h e u s e f u l and induced r o t o r dragl i f t power and inflow; however, it behaves s i m i l a r l y t o t h e l o n g i t u d i n a l f l a p p i n g a n g l e al. An e x p r e s s i o n f o r a' which i n c l u d e s t h e i n f l u e n c e due t o body rate, as d i s c u s s e d i n r e f e r e n c e 8 and i n unpublished Sikorsky A i r c r a f t data, is as follows:

aI =

1

1 - - L12

2BZ

(35)

The e x p r e s s i o n f o r t h e r o t o r torque which accounts f o r b o t h a c c e l e r a t i o n and d e c e l e r a t i o n t o r q u e s i s d e r i v e d from e q u a t i o n s (9) and (11) of r e f e r e n c e 4 and unpublished d a t a o b t a i n e d from S i k o r s k y A i r c r a f t . The t o r q u e c o e f f i c i e n t
may be e x p r e s s e d as a polynomial i n 1-1 times t h e major v a r i a b l e s a s

22

- - - - - cQ 0

=

(0.00io9

0.0036A 0 . 0 0 2 7 e . ~ ~ 1 . 1 0 ~ ~0 . 5 4 5 ~+ ~0 .~1 2 2 02. ~ ~ )

- - - - + (o.ooio9 0 . 0 0 2 7 e . ~ ~ 3 ~ 3 ~ 6.235~e.,, - - - 0 . 1 3 3 x e ~ ~3~+V( - 0 . 9 7 6 ~ ~ 6 . 3 8 ~ 5~.26e~:75)p4

(36)

Thus ,

The t o r q u e a c t i n g on t h e main r o t o r Q,

i s c a l c u l a t e d by using main-rotor

parameters i n e q u a t i o n s (36) and ( 3 7 ) . The r e a c t i o n t o r q u e on t h e f u s e l a g e ,

which i s a f u n c t i o n o f Q,, i s c a l c u l a t e d i n t h e engine dynamics and governor

model d i s c u s s e d i n t h e i s c a l c u l a t e d by using

next tail-

sect roto

io r

n. va

r

i

The t o ables

r i

que act n equat

i i

n o

g o ns

n th (36)

e ta and

i

l (

rot 37).

o

r I

t%1 st

assumed that t h e t a i l - r o t o r r e a c t i o n t o r q u e a c t s on the f u s e l a g e d i r e c t l y and

i s equal t o Qat.

The r o t o r s i d e f o r c e J i n c o n t r o l axes i s c a l c u l a t e d from t h e f o l l o w i n g

e x p r e s s i o n d e r i v e d from e q u a t i o n ( 3 ) i n r e f e r e n c e 3, assuming t h a t terms

el i n v o l v i n g p i t c h and r o l l r a t e may be n e g l e c t e d and t h a t t h e b l a d e p i t c h 8,

can be r e p l a c e d by 0.75 and t e r m s i n v o l v i n g

dropped:

CY

- - blX -23 a01.1x + 1 alblp a0a 1p2 + -61 aOal

- (i - 1 pao - bl

-12

2 bl)e.74

from which

The r o t o r f o r c e s i n c o n t r o l axes are r e s o l v e d i n t o s h a f t a x e s by assuming t h a t t h e components of t h r u s t a l o n g t h e xs- and ys-axes ( f i g . 3 (b) 1 are T B ~ ~ and TAiC, r e s p e c t i v e l y , and t h a t t h e components of drag H and s i d e f o r c e J
along t h e zs-axis are n e g l i g i b l e compared with t h e t h r u s t , so t h a t

' 1 - x = -H cos P J s i n P + mIC r,s

YT I S = -H s i n + J cos + T A ; ~

(40)

23

These f o r c e s are expressed i n h e l i c o p t e r body axes by using t h e transformation

The hub moments due t o f l a p p i n g h i n g e o f f s e t s are given i n s h a f t a x e s by t h e following approximate expressions derived f r o m reference 3, where higher order t e r m s have been neglected:

1 2Lhub,s = ebn V1s

(42)

where t h e c y c l i c f l a p p i n g a n g l e s r e l a t i v e t o s h a f t axes are given by

- a1s = al cos f3 + bl s i n f3 BIC

and

(43)

9 - bls = bl cos f3 al s i n f3 + AIC

The t o r q u e a b o u t t h e r o t o r s h a f t i s g i v e n by

Nhub,s = Qs

(44)

For t h e main r o t o r , Qs i s e q u a l t o Qeng, t h e engine t o r q u e computed i n t h e

s e c t i o n "Engine Dynamics and Governor Model." For t h e t a i l r o t o r , Qs i s

equal to t r a n s form

aQtBi otn

These hub moments a r e e x p r e s s e d i n h e l i c o p t e r body a x e s by t h e

24

The t o t a l moments a c t i n g on the f u s e l a g e due t o t h e main r o t o r and t a i l r o t o r are c a l c u l a t e d by using t h e rotor forces given by equation (41) and hub moments given by e q u a t i o n (45) as

The mathematical model developed h e r e assumes no 6,, h i n g e s on the main

r o t o r , t h a t is, blade-coning and lead-lag motion does n o t a f f e c t blade pitch.

I t i s assumed, however, t h a t t h e t a i l r o t o r h a s l a r g e values o f 63t; f o r example, i f 63t = 4501 t h e n lo o f i n c r e a s e d coning reduces t h e b l a d e p i t c h by

lo and conversely. T h i s e f f e c t i s modeled as f o l l o w s : t h e t a i l - r o t o r coning

angl tive

e t

a

ai lo-tr

o

is tor

c

a c

lc ol

ul le

a c

t t

ed us ive p

ing e itch

quat
Bot.

i

o

n (29) w The new

ith va

l

the c ue of

u

rren
Bot

t

value o f effeci s determined by

solving the first-order differential equation

eo, where

i s t h e c o l l e c t i v e p i t c h v a l u e commanded by t h e p i l o t ' s p e d a l s and

t h e automatic c o n t r o l system. The t i m e c o n s t a n t T63t i s t a k e n as s m a l l as

p o s s i b l e while maintaining a good margin of numerical s t a b i l i t y , A value

between 0.05 and 0.2 second i s t y p i c a l . This method i s s u p e r i o r t o a p u r e l y

eo, a l g e b r a i c c a l c u l a t i o n of

i n t h a t t h e algebraic method w a s found t o be

numerically unstable i n d i g i t a l simulations.

Engine Dynamics and Governor Model
The engine dynamics and governor model w a s adapted from one used by Boeing V e r t o l Company i n a s i m u l a t i o n o f a h e a v y - l i f t h e l i c o p t e r . This model i n c l u d e s the e f f e c t s o f a g a s g e n e r a t o r , a power t u r b i n e , r o t o r i n e r t i a , and s h a f t comp l i a n c e . The f o l l o w i n g d i f f e r e n t i a l e q u a t i o n s d e f i n e t h i s model:
k - - = [Qeng Qam + Kdgov(Qpt 'mj] k m r

25

The i n p u t variables t o t h e s e d i f f e r e n t i a l e q u a t i o n s are Ro, t h e r e f e r e n c e r o t o r

speed, and Qam, t h e aerodynamic t o r q u e a c t i n g on t h e main r o t o r , g i v e n by equa-

4 . t i o n ( 3 7 ) w i t h t h e a p p r o p r i a t e parameters f o r t h e main rotor. The o u t p u t s are

the and

rotor speed

and the

the gas-generator torque

engine Q

torque Qen are interna?

The power-turbine speed v a r i a b l e s . The c o n s t a n t

%R P t

r e p r e s e n t s t h e s h a f t compliancegenThe c o n s t a n t Kdgov i s needed f o r computa-

g?z t i o n a l s t a b i l i t y . The c o n s t a n t s K

and G

are the power-turbine and

gas sta tio

-g nt n

e . t

nerator speed governor g a i n sg',d: The aerodynamic t o r q u e Q,
o a l l o w the governor t o hold re

a p p eTaernsgi asonably

n c

the gas-gen

the onst

a

nQtgreon

t

d or

i

erator t i m e conf f e r e n t i a l equaspeed under wide

l

y

varying aerodynamic torques.

Fuse1age Aerodynamics
The h e l i c o p t e r f u s e l a g e f o r c e and moment data are assumed t o be g i v e n i n equation and table form i n wind-tunnel axes i n t e r m s of local angle o f a t t a c k , l o c a l angle of incidence a t t h e t a i l , angle of s i d e s l i p , body angular rates, and dynamic pressure.
The free-stream angle o f a t t a c k i s g i v e n by

- af = tan-' Was,h Uas,h

(-Tr

=< af

<
= Tr)

(49)

where t h e v e l o c i t y components are c a l c u l a t e d by u s i n g e q u a t i o n s ( 2 0 ) . The free-stream dynamic pressure i s given by

- -1 2
qh - pvas,h

(50)

where

'as, h = \I.:s,h -t- Vas,h + w a2 s , h

The e f f e c t o f t h e main-rotor downwash on t h e l o c a l a n g l e of a t t a c k i s accounted f o r by i n t r o d u c i n g a r o t o r downwash f a c t o r o b t a i n e d from unpublished Sikorsky A i r c r a f t data as

where hm and pm are t h e main-rotor inflow and t i p - s p e e d r a t i o s , respect i v e l y , and Cm i s t h e main-rotor t h r u s t c o e f f i c i e n t . The local a n g l e o f a t t a c k f o r t h e f u s e l a g e i s given by
26

ClfR - af

- emrekf

(-7r =< a f R=< T r )

and t h e l o c a l incidence a t t h e t a i l i s given by

(53)

where ekt and e are empirical c o n s t a n t s which have been determined b y Sikorsky A i r c r a f t $,om wind-tunnel and f l i g h t - t e s t correlation. The c o n s t a n t i t 0 i s t h e f i x e d incidence of the h o r i z o n t a l - t a i l surface.
The f u s e l a g e s i d e s l i p a n g l e i s g i v e n by

Bf = s i n-1 Vas,h
'as,h

(55)

and t h e wind-tunnel yaw a n g l e i s g i v e n by

+w=t 'Bf
Since wind-tunnel d a t a g e n e r a l l y do n o t cover the f u l l ranges of angle of a t t a c k and s i d e s l i p , i t i s assumed t h a t f o r c e and moment c o e f f i c i e n t s remain c o n s t a n t beyond t h e limits of t h e s e angles. This assumption i s based on t h e f a c t t h a t g e n e r a l l y when t h e s e a n g l e s are l a r g e , t h e a i r s p e e d i s low, so t h a t t h e f u s e l a g e f o r c e s and moments are r e l a t i v e l y s m a l l .
The f o r c e s i n wind-tunnel s t a b i l i t y axes axe transformed i n t o body a x e s by the relation

The b a s i c f u s e l a g e aerodynamic moments are assumed t o be g i v e n i n body axes, and t h e t o t a l f u s e l a g e aerodynamic moments, i n c l u d i n g t h e e f f e c t s of the 6.9. being o f f s e t from t h e wind-tunnel mounting p o i n t as w e l l as damping due t o a n g u l a r v e l o c i t y and r o t o r downwash, are o b t a i n e d as follows:
27

- L f , h = %t 'f,hYwt,h 'f,hzwt,h +

-

M f , h = %t + 'f,hzwt,h

'f,hxwt,h

+- Ld,h

Md,h KfeTm

+

+

- N f , h = N w t + 'f,hxwt,h

'f,hYwt,h + Nd,h

The l a s t tern i n t h e Mf,h e q u a t i o n r e p r e s e n t s the s t a t i c moment due t o mainr o t o r downwash a t t h e h o r i z o n t a l t a i l . The terms Ld,h, Ma,@, and Nd,h account f o r the aerodynamic moments due t o body a n g u l a r v e l o c i t i e s and are d i s cussed s u b s e q u e n t l y i n " A p p l i c a t i o n t o U,S. Army CH-54 H e l i c o p t e r and Cargo Container. I'

E x t e r n a l - Load Aerodynamics
The aerodynamic f o r c e s and moments a c t i n g on t h e e x t e r n a l l o a d are calcul a t e d i n e s s e n t i a l l y t h e s a m e way as t h e f u s e l a g e aerodynamics: i n t e r m s of dynamic p r e s s u r e , a n g l e s of a t t a c k and s i d e s l i p , and body a n g u l a r rates. The r o t o r downwash e f f e c t s are n o t considered i m p o r t a n t f o r l o n g c a b l e s as f a r as pendulum and r o c k i n g motions are concerned, b u t e x p e r i e n c e h a s shown t h a t some s i m u l a t e d downwash i s n e c e s s a r y t o provide aerodynamic damping of v e r t i c a l bounce-type o s c i l l a t i o n s near hover due t o cable e l a s t i c i t y . Generally force and moment d a t a are missing a t l a r g e a n g l e s o f a t t a c k , s o t h a t some t y p e of t r i g o n o m e t r i c f o r m u l a t i o n ' f o r forces and moments i s r e q u i r e d . Specific d e t a i l s of the aerodynamic c h a r a c t e r i s t i c s f o r a p a r t i c u l a r e x t e r n a l load are discussed subsequently i n the application section.

Load-Ground Contact Model
Since t h e pickup and release of s l i n g l o a d s are t o be s t u d i e d , a mathematical model of t h e load-ground c o n t a c t f o r c e s i s r e q u i r e d , This model i s d e r i v e d on t h e assumptions t h a t (1) t h e ground can be r e p r e s e n t e d by s p r i n g s and d a s h p o t s , so t h a t v e r t i c a l f o r c e s a c t on each c o r n e r of t h e base of t h e l o a d i n p r o p o r t i o n t o t h e d i s t a n c e t h a t t h e c o r n e r is below ground l e v e l , and (2) viscous s l i d i n g f r i c t i o n r e t a r d s t r a n s l a t i o n a l motion.
The i n e r t i a l p o s i t i o n of the i t h c o n t a c t p o i n t i s determined from t h e relations

28

where t h e f i r s t term on t h e right-hand s i d e d e f i n e s t h e i n e r t i a l p o s i t i o n of t h e l o a d c.g.; t h e second t e r m , t h e c o n t a c t p o i n t with r e s p e c t t o t h e c.g. The i n e r t i a l v e l o c i t i e s of t h i s load c o n t a c t p o i n t a r e given by

The v e r t i c a l ground c o n t a c t f o r c e s a r e computed f o r each c o r n e r o f t h e load by the equation

'ci e = -K p c i , e - K v w c i , e

(i = 1, 2 , 3, 4)

(61)

If Zcice > 0, then t h e load i s being pulled toward the Earth; therefore,
'ci ,e is set t o zero. The v i s c o u s s l i d i n g f o r c e s a r e then determined i n
terms of t h e v e r t i c a l force and t h e h o r i z o n t a l component of v e l o c i t y as

and

f o r i = I, 2 , 3, 4. This model assumes t h a t t h e s l i d i n g f o r c e i s propor-

tional to
vo = 0.31

the v e r t i c a l force Zci,e m/sec and p r o p o r t i o n a l

to

for the

sliding product

velocities greater than of v e r t i c a l f o r c e and

s l i d i n g v e l o c i t y f o r v e l o c i t i e s l e s s than t h i s value. The l o a d c o n t a c t

f o r c e s a t t h e i t h c o n t a c t p o i n t are transformed i n t o load body axes by t h e

transformation

29

'ci ,e 'ci,e .%ie,

(i = 1, 2, 3 , 4)

(64)

and t h e moments a c t i n g on t h e l o a d due t o ground c o n t a c t forces a t t h e i t h contact point are

The t o t a l f o r c e s and moments a c t i n g on t h e load a r e t h u s given by

and

( i = 1, 2 , 3, 4)

(66)

Load Suspension System
The mathematical model f o r t h e e x t e r n a l - l o a d suspension s y s t e m i s based on t h e assumption t h a t t h e cable(s) may be r e p r e s e n t e d by s p r i n g ( s 1 without damping. This assumption allows the cable tension t o be c a l c u l a t e d e a s i l y i n t e r m s of the cable spring constant and the vector d i s t a n c e between t h e cable attachment p o i n t s . With a d i g i t a l simulation t h e i n e r t i a l p o s i t i o n s of t h e cable attachment points are readily calculated, thus simplifying t h e calculat i o n o f the d i s t a n c e between t h e p o i n t s , I t w a s found by t r i a l and e r r o r t h a t
30

t h e simulated cable s p r i n g c o n s t a n t should be s e l e c t e d such that the cable v e r t i c a l bounce frequency i s n o t h i g h e r t h a n approximately 2 Hz, i n o r d e r t o maintain numerical s t a b i l i t y w i t h an i n t e g r a t i o n step s i z e of 1/32 second i n digital simulations.
The i n e r t i a l p o s i t i o n of the cable attachment p o i n t on t h e h e l i c o p t e r shown i n f i g u r e 4 i s c a l c u l a t e d i n terms of the i n e r t i a l p o s i t i o n of t h e h e l i copter 6.9. and t h e d i s t a n c e between t h i s p o s i t i o n and t h e attachment p o i n t as

The i n e r t i a l p o s i t i o n of t h e cable attachment p o i n t on t h e l o a d shown i n f i g u r e 4 i s c a l c u l a t e d s i m i l a r l y as

The c a b l e l e n g t h i s determined f r o m t h e s q u a r e r o o t of t h e sum o f the s q u a r e s of d i f f e r e n c e s i n i n e r t i a l coordinates of cable attachment p o i n t s as

J - - ' - RC = ( xha,e xRa,e)2 + (Yha,e y!La,e>2 ('ha,e

'Ra,e) 2

If t h e unstretched cable length i s Rco, then t h e cable tension is simply

where Ksc i s t h e cable s p r i n g c o n s t a n t . A t e s t on t h e sign o f Tc i s cont i n u o u s l y made t o i n s u r e t h a t t h e cable does n o t e x e r t a compression f o r c e .
- I f (Rc RcO) becomes n e g a t i v e , t h e n Tc i s set t o zero.
The cable t e n s i o n f o r c e must be e x p r e s s e d i n h e l i c o p t e r and l o a d body axes so t h a t i t s e f f e c t can be included i n t h e equations o f motion. Before t h e
31

- - t e n s i o n f o r c e can be expressed i n body axes, t h e i n e r t i a l o r i e n t a t i o n of t h e
cable assumed s t r a i g h t must be determined.
The a n g l e t h a t t h e cable makes w i t h t h e v e r t i c a l i n t h e n o r t h - v e r t i c a l p l a n e i s d e f i n e d i n f i g u r e 4 and i s given by

- = tm-1 XRa,e xha,e

Bc,e

'Ra,e - 2h a , e

The i n e r t i a l c a b l e angle i n t h e east-west d i r e c t i o n i s a l s o d e f i n e d i n f i g u r e 4 and i s given by

Three cable d i r e c t i o n cosines a r e defined with respect t o Earth axes by

- d l c , e

= sin

Bc , e

cos

ac , e

-
-

(XRa,e

Xha,e)/Rc

- -
d2c,e = -sin ac,e - (YRa,e Yha,e)/Rc

- d3c,e = cos 6C r e cos ac , e -- ('Ra,e

'ha,e)/'c

(73)

The t e n s i o n f o r c e i s r e s o l v e d i n t o h e l i c o p t e r body a x e s by t h e e x p r e s s i o n

The moments a c t i n g on t h e h e l i c o p t e r due t o t h e c a b l e t e n s i o n are computed according to

1- -
Lt , h - 't,hYaIh 't,hza,h

Mt,h = 't,hza,h Nt , h = 't,hxa,h

- 't,hxa,h - 't ,hya, h

32

The t e n s i o n f o r c e i s r e s o l v e d i n t o l o a d body a x e s by t h e expression and t h e moments a c t i n g on t h e l o a d due t o t h e cable t e n s i o n are given by

Equations of Motion
The e q u a t i o n s of motion f o r b o t h t h e h e l i c o p t e r and t h e e x t e r n a l s l i n g l o a d are developed i n body axes w i t h r e s p e c t t o a f l a t , n o n r o t a t i n g E a r t h . It i s assumed f o r convenience t h a t each body is r i g i d and t h a t t h e Xh'Zh plane and t h e XR-ZR p l a n e are planes of m a s s symmetry and t h a t gyroscopic e f f e c t s o f e n g i n e s a r e n e g l i g i b l e . The e q u a t i o n s o f motion f o r t h e h e l i c o p t e r are developed f i r s t .
The t r a n s l a t i o n a l motion e q u a t i o n s f o r t h e h e l i c o p t e r are g i v e n i n terms of body-axes components of angular v e l o c i t y , t r a n s l a t i o n a l v e l o c i t y , accelerat i o n s , and components of g r a v i t y as

where

ax,h -- CFx,h/mh
aYth = cFy,h/mh
az ,h - CFz,h/%

(79) 33

The summations i n c l u d e a l l e x t e r n a l f o r c e s : main- and t a i l - r o t o r f o r c e s , fusel a g e aerodynamic f o r c e s , and cable suspension f o r c e s . The l a s t terms on t h e right-hand s i d e of equations (78) give t h e a c c e l e r a t i o n components due t o g r a v i t y . The i n e r t i a l v e l o c i t y o f t h e h e l i c o p t e r i s g i v e n i n body a x e s by i n t e g r a t i n g e q u a t i o n s (78) as

=
Uc9,h

;cg,h

1 Vcg,h =

'cg,h

d t + ucg,h(')
} d t + vc9,h (0)

(80)

The e q u a t i o n s of r o t a t i o n a l motion a r e used i n t h e f o l l o w i n g form:

ph = k L h - ('zz,h - lyy,h)qhrh + 'xz,hphqh

7

+ h.'[

- ('yylh - 'xx,h)Phqh - ' x z l h ~ h r h'x]z~ )h/ ( ' x x l h - 'x2z,h/'zz,h)
zz,h

The body-axes components o f a n g u l a r v e l o c i t y a r e determined by i n t e g r a t i n g equations (81) as
sph = ph d t + Ph(O) sqh = Gh d t + qh ('1
1. r h = rh d t + r h (0)
The h e l i c o p t e r E u l e r a n g l e s shown i n f i g u r e 3 ( a ) are determined by integrating the following differential equations :
34

The i n e r t i a l v e l o c i t y of t h e h e l i c o p t e r c.g. expressed i n body axes i s given i n Earth coordinates by t h e transformation
The Earth-axes c o o r d i n a t e s of t h e h e l i c o p t e r c - g . are determined by i n t e g r a t i n g equation (84) to obtain
The e q u a t i o n s of motion f o r t h e e x t e r n a l l o a d may be o b t a i n e d by changing
a l l t h e s u b s c r i p t s h t o k i n e q u a t i o n s (78) t o ( 8 5 ) . The e x t e r n a l - l o a d
e q u a t i o n s of motion t h u s o b t a i n e d may then be s o l v e d t o g e t h e r w i t h t h e equations of motion f o r t h e helicopter to obtain load motions.
APPLICATION TO U.S. ARMY CH-54 HELICOPTER AND CARGO CONTAINER Rotor and f u s e l a g e d a t a f o r t h e U.S. Army CH-54 h e l i c o p t e r were o b t a i n e d from unpublished Sikorsky A i r c r a f t d a t a , f l i g h t tests a t Langley Research Center, and r e f e r e n c e 9. The f u s e l a g e and main- and t a i l - r o t o r d a t a used i n t h e s i m u l a t i o n are l i s t e d i n t a b l e I. The f u s e l a g e wind-tunnel d a t a are g i v e n i n f i g u r e s 5 t o 11. These c u r v e s are e n t e r e d w i t h l o c a l f u s e l a g e a n g l e o f
a t t a c k a f g l wind-tunnel yaw ( s i d e s l i p ) a n g l e $wt, l o c a l i n c i d e n c e a t t h e
t a i l it, and dynamic pressure Gh as determined from e q u a t i o n s (53) , (56) ,
(54), and ( 5 0 ) , r e s p e c t i v e l y . Values of ekt and ekf of 1.8 and 0 . 5 , respectively, w e r e used i n these equations t o determine fuselage l i f t , s i d e f o r c e , r o l l i n g moment, p i t c h i n g moment, and yawing moment.
35

The f u s e l a g e l i f t i s c a l c u l a t e d by u s i n g t h e d a t a from f i g u r e s 5 and 6 and the following equation :

The f u s e l a g e wind-tunnel p i t c h i n g moment i s c a l c u l a t e d by u s i n g t h e d a t a from f i g u r e s 9 and 10 and t h e following equation:

(87)

The f u s e l a g e d r a g i s c a l c u l a t e d a c c o r d i n g to t h e formula o b t a i n e d f r o m Sikorsky Aircraft
,.Dh = (7.25 -f- 2 . 4 a f R + 42.9af2R -f- 4 5 . 6 $21 ~ ~ ) : ~

where a f R and $wt are i n r a d i a n s .
The f u s e l a g e moments due t o body a n g u l a r rates a r e computed a c c o r d i n g t o t h e formulas obtained from unpublished Sikorsky A i r c r a f t d a t a a s

Ld,h = 95-6rhVas,h Md,h = -218qhvas,h Nd,h = -322rhVas,h

(89)

where Vas,h i s computed by u s i n g equation ( 5 1 ) , and rh and qh are i n

r a d i a n s p e r second. N o t e t h a t t h e r o l l i n g moment Ld,h i s a f u n c t i o n of rh

n o t ph. The f u s e l a g e aerodynamic f o r c e s and moments t h u s determined are used

i n equations (58). It is noted t h a t i n the transformation o f aerodynamic

f o r c e s from wind-tunnel t o body axes given by equation ( 5 7 ) , t h e s i d e s l i p

& a n g l e

w a s i n a d v e r t e n t l y set t o zero. The change i n h a n d l i n g q u a l i t i e s

caused by t h i s error i s considered t o be n e g l i g i b l e a t cruise c o n d i t i o n s and

nonexistent a t hover.

Mass d a t a , c a b l e attachment p o i n t c o o r d i n a t e s , ground c o n t a c t c o o r d i n a t e s and p a r a m e t e r s , and nominal c a b l e l e n g t h used f o r a 2.4-m by 2.4-m by 6.1-m c a r g o c o n t a i n e r are given i n t a b l e 11. Aerodynamic d a t a used f o r t h e c a r g o c o n t a i n e r are d e r i v e d below from d a t a given i n r e f e r e n c e s 10 and 11. Since l a r g e a n g l e s of s i d e s l i p and a n g l e s of a t t a c k w e r e expected during c r u i s e and
hover f l i g h t , t h e wind-tunnel d a t a - which ranged from -5O t o 45O i n p i t c h and

36

from Oo t o 95O i n yaw - had t o be modified t o a l l o w +_180° angle-of-attack v a r i a -
t i o n and +90° a n g l e - o f - s i d e s l i p v a r i a t i o n . This w a s accomplished by f i t t i n g t r i g o n o m e t r i c f u n c t i o n s t o t h e e x i s t i n g d a t a , as shown i n e q u a t i o n s (92) and (94).

The l o a d components of f r e e - s t r e a m v e l o c i t y uas,R' Vas,Rl and

are

obtained by adding steady-wind, g u s t , and i n e r t i a l components c a l c u l a t e d by

using the load version of equations (4), ( l o ) , and (80) as follows:

Uas,R = ucg,R + Ugust,R + Uwind,R
Vas,R = vc g , ~ V g u s t,R + Vwind,R +

Wa s , R =

W

c

g

r ~ Wg u s t , % +

-I-Wwind,R

Then t h e l o a d a n g l e s o f a t t a c k and s i d e s l i p and t h e dynamic p r e s s u r e a r e c a l c u l a t e d from t h e following expressions:

- CXR = tan- 1 Uas,R

62

=

s

i

-1 n

- V as,R

'as ,R

-
'as,&

2

2

A2

U as,R + Vas,R +

I

It w a s found i n s i m u l a t i o n t h a t d u r i n g hover w i t h no winds, d i v e r g e n t o s c i l l a t i o n s of load angular motions would occur. This w a s due t o t h e f a c t t h a t i n t h e zero-airspeed condition, load o s c i l l a t i o n s due to cable s t r e t c h w e r e essent i a l l y undamped and t h e s l i g h t phase s h i f t due t o t h e numerical i n t e g r a t i o n o f the equations of motion caused a numerical i n s t a b i l i t y . This i n s t a b i l i t y w a s e l i m i n a t e d by adding a c o n s t a n t " r o t o r downwash" v a l u e t o t h e 2%-component o f velocity as follows:

- A
Was,G = was,R

9.14

The c a r g o c o n t a i n e r l i f t , d r a g , and s i d e f o r c e s i n wind a x e s a t t h e l o a d geom e t r i c c e n t e r are c a l c u l a t e d from expressions derived from t h e d a t a i n r e f e r ence 10 as
37

..,
L~

=

(6.5

sin

2al

cos

$,)GR

- - = c20.9 7 . 6 6 ( 1 + cos 2 a ~cos BR)IGg

..,
yR = (-7.9 s i n 28, c o s 2aR);iR

(92)

These f o r c e s are expressed i n l o a d body axes by using t h e transformation

-cos aR c o s PR- -cos all s i n PR

-
s i n a1

- s i n a , cos BR - s i n a , s i n BR -cos a1
The cargo c o n t a i n e r aerodynamic moments are g i v e n below and are ased on t h e assumption t h a t t h e s t a t i c p i t c h i n g and r o l l i n g moments a r e n e g l i g i b l e i n comp a r i s o n w i t h t h e p i t c h i n g and r o l l i n g moments caused by t h e suspension system. The s t a t i c yawing moment was d e r i v e d from t h e d a t a i n r e f e r e n c e 10 a s

v e l o c i t i e s are assumed t o be p r o p o r t i o n a l t o t h e product o f a i r s p e e d and angul a r v e l o c i t y , as i n t h e c a s e of t h e f u s e l a g e moments given by e q u a t i o n s (891, and are given by

where pR, qRl and r, are i n r a d i a n s p e r second. The t o t a l aerodynamic moments a c t i n g on t h e l o a d are t h u s given by
38

SIMULATION DESCRIPTION
Computer Hardware
The mathematical model h a s been programed i n FORTRAN I V f o r o p e r a t i o n on t h e Langley real-time s i m u l a t i o n system u s i n g t h e Control D a t a CYBER 175 d i g i t a l computer system. The program a c c e p t s i n p u t s from t h e s i m u l a t i o n cockp i t through A D C ' s , and o u t p u t s v o l t a g e s t o t h e s i m u l a t o r through DAG'S. The program flow i s c o n t r o l l e d by a n o p e r a t o r through use of a s i m u l a t i o n c o n t r o l console (fig. 12).
The s i m u l a t i o n c o n t r o l c o n s o l e c o n s i s t s o f w h i t e i n d i c a t o r l i g h t s , r e d i n d i c a t o r l i g h t s , f u n c t i o n s e n s e s w i t c h e s , mode c o n t r o l s w i t c h e s , a d a t a e n t r y keyboard, d i g i t a l decimal d i s p l a y u n i t , and p o t e n t i o m e t e r s . The w h i t e i n d i c a t o r l i g h t s are used t o i n d i c a t e program s t a t u s o r d i a g n o s t i c s . The r e d i n d i c a t o r l i g h t s are used t o i n d i c a t e program d i a g n o s t i c s . The f u n c t i o n sense s w i t c h e s are used t o s e l e c t program o p t i o n s . The mode control s w i t c h e s shown i n f i g u r e s 13 and 14(a) are used t o c o n t r o l t h e running of t h e RTS computer program. Each s w i t c h (mode) i s b r i e f l y d e s c r i b e d as t o i t s use (mode nominally a c t i v e when s w i t c h d e p r e s s e d ) :
- OPER (OPERATE) begins u p d a t i n g time and i n t e g r a t i n g t h e d i f f e r e n t i a l
equations
HOLD - h o l d s i n t e g r a t e d v a r i a b l e s a t p r e v i o u s value
RESET - i n i t i a l i z e s program a t Time = 0
IDLE - i d l e s t h e computer (no computations)
CHANGE - changes program v a r i a b l e t o t h e new v a l u e e n t e r e d on t h e d a t a
entry keyboard
- SCAN s c a n s through tables and d i s p l a y s v a l u e s on t h e d i g i t a l decimal
display unit
RELEASE - releases CHANGE and SCAN modes
- E R ~ S E erases r e a l - t i m e d i s k f i l e
TERM (TERMINATE) - terminates program a t s i m u l a t i o n c o n t r o l console and
transfers control to the Tektronix terminal
READ - l o a d s r e a d o v e r l a y
- PRINT l o a d s p r i n t o v e r l a y
RELEASE - releases ERASE, TERM, READ, and PRINT modes
The d a t a e n t r y keyboard shown i n f i g u r e s 1 3 and 1 4 ( b ) i s used t o i n p u t new v a l u e s f o r program v a r i a b l e s . The keyboard i s used i n c o n j u n c t i o n w i t h t h e
39

d i g i t a l decimal d i s p l a y shown i n f i g u r e s 1 3 and 1 4 ( c ) . Any program v a r i a b l e can be changed w i t h t h e s e two u n i t s by f o l l o w i n g a simple procedure.
The p o t e n t i o m e t e r s are used t o i n p u t v a r i a b l e s ( t h r o u g h A D C ' s ) . They are mainly used f o r checkout,
The T e k t r o n i x t e r m i n a l i s used t o communicate i n t e r a c t i v e l y w i t h t h e CYBER 175. When t h e RTS program i s running, t h e t e r m i n a l i s used as an o u t p u t device, f o r example, f o r error messages,
Cockpit
The c o c k p i t , which i s l i n k e d t o t h e CYBER 175 computer f o r t h e s i m u l a t i o n , is shown i n f i g u r e s 15 t o 18, I t h a s s i m u l a t e d i n s t r u m e n t a t i o n , p i l o t c o n t r o l s , and a v i s u a l landing d i s p l a y system t o simulate a h e l i c o p t e r cockpit.
The c o c k p i t i n s t r u m e n t p a n e l i s shown i n f i g u r e 15. F i g u r e 16 shows t h e c o n t r o l s by which p i l o t i n p u t s are f e d i n t o t h e CYBER 175 computer. They include the standard helicopter controls: cyclic stick with t r i m release button and " c o o l i e h a t " t r i m c o n t r o l , c o l l e c t i v e s t i c k , and a n t i t o r q u e pedals.
The c y c l i c s t i c k i s o p e r a t e d by a t h r e e - a x i s h y d r a u l i c c o n t r o l l o a d e r which i s c o n t r o l l e d by t h e a n a l o g computer shown i n f i g u r e 19. S t i c k t r a v e l a t t h e c e n t e r o f t h e hand g r i p i s l i m i t e d by mechanical s t o p s t o 2 1 2 . 7 cm i n t h e l a t e r a l d i r e c t i o n and t14.0 cm i n t h e l o n g i t u d i n a l d i r e c t i o n . The s t i c k dynamics are modeled on t h e analog computer as second-order systems with t h e damping chosen by t h e p i l o t . The s t i c k f o r c e g r a d i e n t used i s 4.4 N/cm i n t h e l o n g i t u d i n a l a x i s and 8.9 N/cm i n t h e l a t e r a l a x i s and w a s o b t a i n e d from r e f e r e n c e 9. The f o r c e t h e p i l o t a p p l i e s t o t h e s t i c k i s opposed by t h e h y d r a u l i c c o n t r o l l o a d e r and measured by f o r c e t r a n s d u c e r s . The f o r c e t r a n s d u c e r s i g n a l s are f e d i n t o t h e a n a l o g computer and t h e s t i c k a c c e l e r a t i o n i s c a l c u l a t e d . The a c c e l e r a t i o n s i g n a l i s processed together with t h e a c t u a l s t i c k p o s i t i o n and a p o s i t i o n error s i g n a l i s formed. This e r r o r s i g n a l i s then s e n t t o t h e c o n t r o l l o a d e r s e r v o which moves t h e s t i c k . The new s t i c k p o s i t i o n i s f e d through an ADC t o t h e CYBER 175 computer a s t h e p i l o t ' s c y c l i c s t i c k i n p u t . The l a t e r a l p o s i t i o n of t h e s t i c k i s t h e i n p u t xlat i n equations ( 1 2 ) and t h e l o n g i t u d i n a l p o s i t i o n i s t h e i n p u t xlon i n e q u a t i o n s ( 1 2 ) .
I n t h e trimming mode, t h e CYBER 175 computer uses t h e t r i m a l g o r i t h m s d i s cussed subsequently i n the section "Trim Calculations" t o obtain the t r i m posit i o n of t h e c y c l i c s t i c k . T h i s p o s i t i o n i s f e d through a DAC t o t h e c o n t r o l l o a d e r analog computer as a command t o d r i v e t h e s t i c k t o t h e t r i m p o s i t i o n .
A c t i v a t i n g t h e t r i m release b u t t o n on t h e c y c l i c s t i c k o r s t i c k t r i m t o g g l e s w i t c h shown i n f i g u r e s 16 and 18 removes t h e t r i m f o r c e . The s t i c k position at the instant the p i l o t releases the button or turns the stick t r i m t o g g l e on becomes t h e new t r i m p o s i t i o n . The t r i m p o s i t i o n may also b e moved by t h e f o u r - p o s i t i o n c o o l i e h a t mentioned p r e v i o u s l y . By moving t h e c o o l i e h a t
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i n a c e r t a i n d i r e c t i o n , t h e s t i c k ' s z e r o f o r c e p o s i t i o n i s moved i n t h a t d i r e c t i o n , The t r i m release b u t t o n , t h e s t i c k t r i m t o g g l e , and t h e c o o l i e h a t are d i s c r e t e i n p u t s t o t h e CYBER 175 computer.

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i s l i m i t e d by mechanical s t o p s t o 35.6 cm. A f r i c t i o n c o l l a r can be a d j u s t e d

t o v a r y t h e s t i c k f r i c t i o n . The t r i m a l g o r i t h m c a l c u l a t e s t h e c o l l e c t i v e - s t i c k

t r i m p o s i t i o n . The p i l o t must then p l a c e t h e c o l l e c t i v e s t i c k i n i t s t r i m

position, which i s evidenced by t h e t u r n i n g o f f of t h e c o l l e c t i v e t r i m l i g h t

shown i n f i g u r e 18.

The p e d a l s are d r i v e n by t h e s a m e t h r e e - a x i s h y d r a u l i c c o n t r o l l o a d e r syst e m which d r i v e s t h e c y c l i c s t i c k ; however, t h e response c h a r a c t e r i s t i c s are modeled on t h e CYBER 175 computer because t h e frequency response of t h e p e d a l s i s n o t as high as t h a t o f t h e c y c l i c s t i c k . The pedal t r a v e l i s l i m i t e d by mechanical stops t o approximately 10 cm. The p e d a l dynamics are second o r d e r w i t h a f o r c e g r a d i e n t o f 62.2 N/cm, as o b t a i n e d from r e f e r e n c e 9. The p e d a l damping w a s chosen by t h e p i l o t . The o p e r a t i o n of the p e d a l c o n t r o l 1oad.er is i d e n t i c a l w i t h t h a t of t h e c y c l i c s t i c k . Microswitches l o c a t e d on each p e d a l have t h e s a m e f u n c t i o n as t h e t r i m r e l e a s e push b u t t o n l o c a t e d on t h e c y c l i c s t i c k . I n t h e trimming mode, t h e t r i m a l g o r i t h m c a l c u l a t e s the t r i m p o s i t i o n f o r t h e p e d a l s . These p o s i t i o n commands are f e d through DAC's t o d r i v e t h e pedals to their t r i m position,

The remaining p i l o t c o n t r o l s i n t h e c o c k p i t are d i s c r e t e i n p u t s t o t h e CYBER 175 computer. They are
Winch c o n t r o l - t h r e e - p o s i t i o n t o g g l e s w i t c h t o l e n g t h e n o r s h o r t e n t h e
load cable
Mode c o n t r o l b u t t o n s - shown i n f i g u r e 18; s e l e c t one o f t h e f o l l o w i n g
modes of o p e r a t i o n :

RESET

resets to i n i t i a l f l i g h t conditions

HOLD

holds present flight conditions

OPERATE

starts simulation

AE'CS s e l e c t o r b u t t o n s - shown i n f i g u r e 18; s e l e c t t h e f o l l o w i n g o p t i o n s :

AFCS

automatic f l i g h t c o n t r o l system on

YAW

heading h o l d on

ALT

a l t i t u d e h o l d on

HOVER

hover hold on

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- S t i c k t r i m s w i t c h two-position t o g g l e switch t o t u r n c y c l i c - s t i c k - f o r c e
t r i m system on and o f f ; shown i n f i g u r e 18
Load release b u t t o n - l o c a t e d on t h e c y c l i c s t i c k ; when p r e s s e d , e x t e r n a l
load is released
Several additional features are included i n the cockpit.
The p i l o t views a v i s u a l t e r r a i n scene provided by t h e v i s u a l l a n d i n g d i s p l a y system (VLDS), and a computer-generated l o a d / l a n d i n g zone scene. P i l o t s ' comments d u r i n g checkout i n d i c a t e d t h a t a s e p a r a t e and independent computerg e n e r a t e d video d i s p l a y of t h e s l i n g l o a d and l a n d i n g zone as viewed by a downlooking TV camera a t t a c h e d t o t h e h e l i c o p t e r a t t h e p i l o t ' s l o c a t i o n is unaccepta b l e . Many hours o f s i m u l a t i o n experience w i t h NASA p i l o t s and p i l o t s f r o m i n d u s t r y have shown t h a t an a c c e p t a b l e s i m u l a t i o n s l i n g l o a d d i s p l a y can be made by e l e c t r o n i c a l l y combining t h e v i s u a l t e r r a i n scene w i t h t h e computerg e n e r a t e d scene. The r e s u l t i n g image i s t h e n d i s p l a y e d on t h e video monitor p l a c e d i n t h e normal e y e - l e v e l p o s i t i o n . F i g u r e 1 7 shows t h e v i r t u a l image l e n s system through which t h e p i l o t views t h e c o l o r monitor which d i s p l a y s t h e v i s u a l t e r r a i n scene e l e c t r o n i c a l l y mixed w i t h t h e down-looking l o a d scene. The v i r t u a l image l e n s c a u s e s t h e p i l o t ' s e y e s t o be focused a t i n f i n i t y . A l s o , shown i n f i g u r e 1 7 i s a s m a l l b l a c k and white monitor which shows t h e same d i s p l a y s a s t h e p i l o t ' s c o l o r monitor. The b l a c k and white monitor i s used by t h e researcher s i t t i n g i n the r i g h t s e a t of the cockpit.
A n audio generator c r e a t e s cockpit n o i s e , with one component having a frequency e q u a l t o 6 t i m e s r o t o r speed ( i n rpm) and t h e o t h e r component having
white n o i s e w i t h magnitude p r o p o r t i o n a l t o t h e q u a n t i t y (VaS,h + 24.4) m/sec
t o r e p r e s e n t wind n o i s e . These audio s i g n a l s are o u t p u t on a speaker i n t h e c o c k p i t w i t h volume l e v e l based on p i l o t comments.
Simulation Software
The s i m u l a t i o n s o f t w a r e i s w r i t t e n i n FORTRAN, w i t h some assembly language. It i s broken down i n t o primary and secondary o v e r l a y s w i t h many s u b r o u t i n e s and general-purpose functions.
Figure 20 shows t h e c e n t r a l memory l a y o u t w i t h t h e v a r i o u s o v e r l a y s . The l a r g e block l a b e l e d ( 0 , O ) r e p r e s e n t s t h e base o v e r l a y , which r e q u i r e s 40 0008 s t o r a g e l o c a t i o n s . It c o n t a i n s a l l v a r i a b l e s which must be communicated between o v e r l a y s , s u b r o u t i n e s r e q u i r e d by more than one o v e r l a y , and some r e a l t i m e system software.
The primary l e v e l o v e r l a y s (l,O), ( 2 , 0 ) , ( 3 , 0 ) , and ( 4 , O ) s h a r e t h e same memory l o c a t i o n s . T h e i r f u n c t i o n s a r e n e c e s s a r i l y mutually e x c l u s i v e , Overl a y (1,O) i n i t i a l i z e s program v a r i a b l e s and i n i t i a l i z e s t h e r e a l - t i m e system. Overlay ( 2 , O ) p r i n t s t r i m sheets o r t i m e h i s t o r i e s . Overlay (3,O) reads i n d a t a f o r v a r i o u s l o a d s . Overlay ( 4 , O ) i s t h e main o v e r l a y and c o n t a i n s t h e real-time loop. I t i s f u r t h e r described below and i n f i g u r e 2 1 ( a ) , Secondary l e v e l overlays ( 4 , l ) t o ( 4 , l O ) are o p t i o n a l l y executed by overlay ( 4 , O ) .
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The secondary o v e r l a y s s h a r e t h e same memory l o c a t i o n s . Overlay ( 4 , l ) executes s t a t i c checks; ( 4 , 2 ) c o n t a i n s t h e load/landing zone d i s p l a y e q u a t i o n s ; (4,3) contains the t r i m algorithm and logic; (4,4) contains preprocessing equat i o n s , i.e., c a l c u l a t i o n s which can be done o u t o f t h e real-time loop f o r a given run; (4,5) checks and p r i n t s o u t function data; (4,6) c a l c u l a t e s stab i l i t y d e r i v a t i v e s ; (4,7) p r i n t s error messages t o t h e console operator; (4,101 c a l c u l a t e s l i n e a r i z a t i o n d e r i v a t i v e s . Of t h e s e e i g h t f u n c t i o n s , o n l y t h e load/landing zone d i s p l a y e q u a t i o n s are needed i n t h e real-time loop. The others are brought i n as necessary between runs.
The real-time o v e r l a y ( o v e r l a y ( 4 , O ) ) i s t h e main loop, w i t h a l l t h e o t h e r o v e r l a y s s u p p o r t i n g it. A g e n e r a l flow diagram o f t h e main l o o p i s shown i n figure 2 1 ( a ) . This real-time loop c o n s i s t s of three separate loops: r e s e t , hold, and o p e r a t e . The r e s e t l o o p i s c y c l e d through a t T i m e = 0. The h o l d loop i s s e l e c t e d a t any t i m e t o h o l d a l l s i m u l a t i o n v a r i a b l e s f i x e d . The operate loop begins the simulation, calculating and i n t e g r a t i n g the equations of motion and updating t h e independent v a r i a b l e t i m e i n synchronization w i t h real t i m e , with a s t e p s i z e At of 1/32 sec. This r a t e w a s s e l e c t e d a s t h e l a r g e s t t i m e i n t e r v a l allowed by t h e real-time system which g i v e s a c c u r a t e r e s u l t s and updates c o c k p i t i n s t r u m e n t s w i t h no v i s i b l e jumps.
The block i n f i g u r e 2 1 ( a ) l a b e l e d (A) r e p r e s e n t s t h e s e c t i o n of t h e reset loop i n which various secondary o v e r l a y s can be s e l e c t e d and executed. For example, when t h e c o n s o l e o p e r a t o r d e p r e s s e s t h e t r i m b u t t o n , o v e r l a y ( 4 , 4 ) i s executed t o c a l c u l a t e preprocessing equations, and then overlay ( 4 , 3 ) is loaded t o execute t h e t r i m algorithm. Overlay ( 4 , 3 ) w i l l s t a y i n memory u n t i l another option is selected.
Blocks (B) t o ( E ) are cycled through i n b o t h h o l d and reset modes. Block (B) r e p r e s e n t s t h e sampling of 5 analog s i g n a l s ( v i a A D C ' s ) and 2 7 d i s c r e t e i n p u t s from t h e cockpit. These voltages are scaled and t h e r e s u l t i n g parameters (e.g., s t i c k inputs) saved f o r later use.
Block (C) r e p r e s e n t s a l l t h e c a l c u l a t i o n s necessary t o compute t h e equations of motion f o r the helicopter and, optionally, f o r the load. This block i s broken down i n f i g u r e 2 1 ( b ) , where each b l o c k r e f e r s t o a p o r t i o n of t h e p r e v i o u s l y developed mathematical model. Some o f t h e general-purpose a l g o rithms used i n t h i s s e c t i o n a r e a c o n v o l u t i o n i n t e g r a t i o n scheme used t o repres e n t f i r s t - and second-order f i l t e r s , a l i n e a r i n t e r p o l a t i o n scheme f o r funct i o n d a t a t a b l e look-ups f o r f u n c t i o n s of one o r two v a r i a b l e s , and an E u l e r i n t e g r a t o r f o r u s e when speed i s d e s i r a b l e and accuracy n o t c r i t i c a l .
Block ( D ) r e p r e s e n t s t h e c a l c u l a t i o n s f o r t h e VLDS and Adage d i s p l a y s . The VLDS d r i v e e q u a t i o n s are computed, 19 a n a l o g s i g n a l s are o u t p u t , and 11 a n a l o g e r r o r s i g n a l s are f e d back. The Adage d i s p l a y e q u a t i o n s are comp u t e d , and t h e d a t a are t r a n s m i t t e d i n d i g i t a l form t o t h e Adage Graphics Terminal (AGT 1 3 0 ) . F u r t h e r d e s c r i p t i o n s o f t h e VLDS and Adage d i s p l a y s are given be l o w .)
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Block (E) represents t h e s c a l i n g of program v a r i a b l e s and the o u t p u t t i n g of those v a r i a b l e s v i a DAC’s and d i s c r e t e channels. Twenty-nine voltages and 12 discrete outputs are trunked t o the simulator cockpit.
Blocks (F) and ( G ) are o n l y c y c l e d through i n t h e o p e r a t e mode. D a t a f o r
t h e t i m e - h i s t o r y p r i n t o u t are recorded i n b l o c k (F), e v e r y s p e c i f i e d number of
i t e r a t i o n s . These d a t a can be p r i n t e d later a t t h e r e s e a r c h e r ’ s option.
Block ( G ) r e p r e s e n t s t h e i n t e g r a t i o n o f t h e e q u a t i o n s o f motion. The real-time loop i s designed t o be as f a s t a$ p o s s i b l e , w h i l e m a i n t a i n i n g good
accuracy. The Adams-Bashforth second-order (AB-2) , one-pass scheme i s very
f a s t , and i s a c c u r a t e enough f o r t h i s p a r t i c u l a r s i m u l a t i o n , as w a s shown i n t h e comparison of t i m e - h i s t o r y p l o t s w i t h t h e independent check program. The AB-2 i n t e g r a t i o n scheme i s f u l l y d e s c r i b e d i n r e f e r e n c e 1 2 .
The complete real-time l o o p r e q u i r e s a minimum of 2.5 m s e c ( i n reset) and a maximum o f 6.2 msec ( i n o p e r a t e ) . The 2.5-msec f i g u r e i n c l u d e s o n l y t h e h e l i c o p t e r with AFCS, whereas t h e 6.2-msec f i g u r e i n c l u d e s t h e 2.4-m by 2.4-m by 6.1-m l o a d , winds, g u s t , and so f o r t h . Table I11 shows a complete breakdown of the required t i m e .
Visual Landing Display System
The v i s u a l l a n d i n g d i s p l a y system (VLDS) c o n s i s t s of a f i x e d , coloredt e r r a i n board system and a movable camera t r a n s p o r t and i s designed f o r use with a monitor and v i r t u a l image l e n s system f o r d i s p l a y i n g an “out-the-window” scene i n a s i m u l a t e d c o c k p i t . A b r i e f d e s c r i p t i o n follows. A more d e t a i l e d description i s given i n reference 13.
The 7.3-m by 18.3-m t e r r a i n model board of t h e VLDS i n c l u d e s two a i r p o r t s and surrounding t e r r a i n , one a t 750/1 s c a l e and t h e o t h e r a t 1500/1 s c a l e , and i s shown i n f i g u r e 22. There are a t o t a l o f f i v e paved runways, from 0.6 km t o 3.5 km i n l e n g t h . A h e l i p a d i s l o c a t e d on t h e 750/1 a i r p o r t and i s shown i n f i g u r e 23. I t c o n s i s t s of a Maltese c r o s s w i t h a 45-m by 45-m b o r d e r . The t e r r a i n i s g e n e r a l l y f l a t , and p r o v i s i o n i s made f o r variable v i s i b i l i t y , v a r i a b l e cloud-base heights, and day, dusk, and n i g h t scenes.
The camera system has a f i e l d of view 48O wide and 36O h i g h and u s e s a 525-line color video raster system. This system provides nominal resolution on t h e o r d e r of 9 minutes of a r c . The p i l o t ’ s eye p o s i t i o n and t h e o r i e n t a t i o n of t h e p i l o t ’ s l i n e of s i g h t o u t t h e forward window are b o t h c a l c u l a t e d w i t h r e s p e c t t o t h e simulated runway. These p o s i t i o n s and o r i e n t a t i o n a n g l e s (and t h e i r r a t e s ) are used t o d r i v e t h e camera system. The dynamic c h a r a c t e r i s t i c s of t h e VLDS t r a n s p o r t system are given i n table IV.
Load/Landing Zone V i s u a l Display
The s i m u l a t i o n i s i n t e n d e d t o be used by p i l o t s t o e v a l u a t e s l i n g l o a d s t a b i l i z a t i o n systems, and since s l i n g load p i l o t i n g i s a v i s u a l t a s k , a
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r e a l i s t i c d i s p l a y of load motions and landing zone p o s i t i o n must be a v a i l a b l e i n t h e s i m u l a t e d c o c k p i t . P r e l i m i n a r y work with p i l o t s d e f i n e d t h e format of t h e s i m u l a t i o n d i s p l a y as follows. The d i s p l a y should show i n p e r s p e c t i v e t h e three-dimensional o u t l i n e of the external load, the c a b l e ( s ) , and the Earthf i x e d f e a t u r e s such as t h e l a n d i n g zone and l e a d - i n d i s t a n c e marks. The c e n t e r l i n e of t h e d i s p l a y should be l o c a t e d a t t h e cable attachment p o i n t on t h e h e l i copter and t h e view should be s t a b i l i z e d i n p i t c h and r o l l ; t h a t i s , t h e v i e w should always be v e r t i c a l l y o r i e n t e d . Load a l t i t u d e above ground l e v e l should be presented i n analog form by a p o i n t e r and a l t i t u d e scale, and h e l i c o p t e r p i t c h and r o l l a t t i t u d e bars should be generated t o provide the p i l o t p r e c i s e a t t i t u d e information needed while hovering. A sketch of the desired load/ landing zone d i s p l a y i s given i n f i g u r e 24.
The l o a d / l a n d i n g zone computer g r a p h i c d i s p l a y i s d e f i n e d by v e c t o r end p o i n t s . These end p o i n t s are computed by using t h e formulas developed i n equations (97) t o (101).
The c o o r d i n a t e s o f t h e l o a d c.g. are given w i t h r e s p e c t t o h e l i c o p t e r body axes by
The c o o r d i n a t e s o f any p o i n t p on t h e l o a d expressed i n h e l i c o p t e r body a x e s are thus given by
The c o o r d i n a t e s o f t h e h e l i c o p t e r viewpoint e x p r e s s e d i n E a r t h axes are g i v e n by
45

where X v l h l y v l h r Zv,h are t h e c o o r d i n a t e s o f t h e viewpoint i n h e l i c o p t e r axes. Thus, t h e c o o r d i n a t e s o f any p o i n t p on t h e E a r t h w i t h r e s p e c t t o t h e viewpoint are given i n helicopter axes by

where xzp,el Yzp,el zzp,e are t h e c o o r d i n a t e s of any p o i n t p i n E a r t h axes. S i n c e t h e l o a d / l a n d i n g zone d i s p l a y w i l l be shown on a s c r e e n , t h e s e p o i n t s must be e x p r e s s e d i n t h e c o o r d i n a t e system of t h e s c r e e n . The f o l l o w i n g equat i o n s are used t o transform t h e s e p o i n t s t o screen coordinates and t o add perspective :
Xs c r e e n = "p , h p p , h
J Yscreen - yp,h/Zp,h

where %,h' 'plhr 'plh are t h e coordinates a t any p o i n t p i n h e l i c o p t e r axes. The h e l i c o p t e r a t t i t u d e b a r s a r e g e n e r a t e d d i r e c t l y i n s c r e e n c o o r d i n a t e s . The d a t a d e f i n e d i n s c r e e n c o o r d i n a t e s must be c l i p p e d a t t h e s c r e e n boundary. The c l i p p i n g a l g o r i t h m employed i s d e s c r i b e d i n r e f e r e n c e 14.
The l o a d / l a n d i n g zone d i s p l a y requirements are m e t by t h e Adage Graphics Terminal i n c o n j u n c t i o n w i t h t h e CYBER 175 computer. The Adage i s an independent, d i g i t a l , g r a p h i c s computer w i t h an o p e r a t o r console shown i n f i g u r e 25. Computations r e q u i r e d f o r t h e d i s p l a y are performed on t h e CYBER 175 computer and d a t a are t r a n s m i t t e d t o t h e Adage, which g e n e r a t e s t h e d i s p l a y . Sample d i s p l a y s are shown i n f i g u r e s 26 and 27.

Trim Calculations

I n order f o r the simulation t o s t a r t i n an unaccelerated f l i g h t condition,

it is r e q u i r e d t h a t an algorithm t h a t renders c e r t a i n mathematical model t i m e

d e r i v a t i v e s z e r o be a v a i l a b l e f o r determining t h e h e l i c o p t e r a t t i t u d e and con-

t r o l positions and the load a t t i t u d e and position with respect to the h e l i -

copter. The a l g o r i t h m used i s based on t h e s o - c a l l e d method of s e c a n t s and i s

d e s c r i b e d i n appendix D o f r e f e r e n c e 15. The a l g o r i t h m determines t h e v a l u e s

of t h e f o l l o w i n g independent v a r i a b l e s : xlon1 xlat1 xcol, xped1 etfl ohl ohl

Vml V t t

e t ~ t$21 O R , X g c g l e 1 ~

g Z R ~~ ~SO ,~t h~a t t h, e foll~owing ~dependent

46

. v a r i a b l e s are approximately zero:
*m, v t , g o t ' 'cg,R, ;cg,Rp ;cg,R,

. 6 cg,h'

Vcg,h'

*
Wcg,h' phr qh'

rh'

Whcg,e'

42,

F o r t h e s i m u l a t i o n of t h e CH-54 h e l i c o p t e r and cargo c o n t a i n e r t h e algorithm t r i m s a t any s p e c i f i e d a i r s p e e d from -20 t o 100 k n o t s i n approximately 1 sec. Values o f t h e major h e l i c o p t e r v a r i a b l e s i n t r i m a t 0.1, 30, 60, and 90 k n o t s are given i n t a b l e V. The use of 0 . 1 k n o t w a s t o p r e v e n t s i n g u l a r i t i e s t h a t would occur a t e x a c t l y z e r o a i r s p e e d .

VERIFICATION AND VALIDATION
Simulation Verification
The s i m u l a t i o n software v e r i f i c a t i o n w a s accomplished w i t h an independent check program. The check program w a s w r i t t e n i n FORTRAN and w a s designed t o run i n a batch environment. I t w a s developed from t h e mathematical model independently of the real-time program. A s the independent check program i s n o t r e q u i r e d t o run i n r e a l t i m e , a m o r e a c c u r a t e i n t e g r a t i o n a l g o r i t h m t h a n AB-2 w a s selected f o r t h i s use. This procedure provided a verification of both the programing of t h e mathematical model and t h e accuracy of t h e numerical solut i o n . A Runge-Kutta seventh-order, 13-pass i n t e g r a t i o n algorithm with v a r i able s t e p s i z e was used i n t h e independent check program. T h i s a l g o r i t h m and i t s accuracy a r e described i n reference 16.
The check c a s e s e l e c t e d c o n s i s t e d of a 30-sec run, d u r i n g which t h e f o u r c o n t r o l i n p u t s w e r e disturbed independently with a half-cycle sine-wave forcing f u n c t i o n . The AFCS w a s o f f f o r t h e f i r s t 3 s e c , and w a s c u t on and o f f i n 3-sec i n t e r v a l s during the run. This allowed f o r large, b u t reasonable, pert u r b a t i o n s i n t h e s t a t e v a r i a b l e s . The check case w a s run f o r t h e h e l i c o p t e r alone a t hover and a t 60 knots and f o r the h e l i c o p t e r with s l i n g load a t hover and a t 60 knots. The time h i s t o r i e s of t h e s t a t e v a r i a b l e s showed e x c e l l e n t agreement with those from t h e real-time program, and f u r t h e r a n a l y s i s proved t h a t t h e n e g l i g i b l e d i f f e r e n c e s w e r e due t o i n t e g r a t i o n e r r o r s . These e r r o r s were s m a l l enough t o v e r i f y t h a t t h e AB-2 i n t e g r a t i o n scheme w a s s u f f i c i e n t l y accurate for the simulation.
Hardware v e r i f i c a t i o n c o n s i s t s of a s t a t i c check performed b e f o r e each real-time s e s s i o n . The i n s t r u m e n t s are checked v i s u a l l y f o r conformity w i t h s t a t i c check v a l u e s . A l l t h e c o n t r o l i n p u t s are d e f l e c t e d t o t h e i r maximum and minimum p o s i t i o n s while t h e console o p e r a t o r monitors t h e v o l t a g e s . Finally, a l l the discrete inputs are activated while the console operator monitors them.

Mathematical Model V a l i d a t i o n
The h e l i c o p t e r p o r t i o n o f t h e p r e v i o u s l y d e s c r i b e d mathematical model w a s v a l i d a t e d by comparison of simulated t i m e h i s t o r i e s with f l i g h t d a t a , a n a l y s i s of eigenvalues and eigenvectors, and by p i l o t evaluation.

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Eleven s i m u l a t i o n s w e r e made f o r comparison w i t h f l i g h t t i m e h i s t o r i e s , and t h e r e s u l t s from two of t h e s e are shown i n f i g u r e s 28 and 29. I n a l l cases the simulated helicopter m a s s characteristics were set equal to those of the h e l i c o p t e r t h a t generated t h e f l i g h t t i m e h i s t o r i e s . (These f l i g h t s were conducted a t Langley i n 1972 i n an u n r e l a t e d s l t n g l o a d p r o j e c t r e p o r t e d i n r e f . 17.) Table I gives the m a s s c h a r a c t e r i s t i c s and o t h e r d a t a used i n t h e s i m u l a t i o n . The AFCS w a s o f f i n a l l cases. When reviewing f i g u r e s 28 and 29, t h e r e a d e r should keep i n mind t h a t t h e b a s i c h e l i c o p t e r is u n s t a b l e a b o u t each a x i s , so t h a t any perturbation from t r i m w i l l r e s u l t i n divergent r o l l , p i t c h , and yaw motion because of t h e dynamic c r o s s coupling. T h i s divergence t e n d s t o amplify d i f f e r e n c e s between f l i g h t and s i m u l a t i o n as t i m e i n c r e a s e s . I t was the manufacturer's opinion t h a t about 2 o r 3 sec of close agreement between simulation and f l i g h t d a t a would be a l l t h a t should be expected.
Figure 28 shows t h e d a t a f o r a run i n which t h e h e l i c o p t e r alone w a s hovering and t h e p i l o t made a l o n g i t u d i n a l c y c l i c s t e p i n p u t of about 3 c m f o r approximately 2 s e c while h o l d i n g t h e o t h e r c o n t r o l s approximately f i x e d . The s i m u l a t i o n w a s trimmed a t hover and t h e c o n t r o l p o s i t i o n changes made i n f l i g h t w e r e d u p l i c a t e d i n t h e s i m u l a t i o n by u s i n g t h e recorded c o n t r o l i n p u t d a t a . The upper two p l o t s i n f i g u r e 28 show 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 s t i c k motion f o r t h e f i r s t 5 s e c of t h i s case. The p e d a l s moved less than 0.05 cm d u r i n g t h i s run and t h e c o l l e c t i v e s t i c k moved l e s s than 0.09 cm a t t h e g r i p . The d i f f e r e n c e s i n E u l e r a n g l e s and body r a t e s between f l i g h t and s i m u l a t i o n a t t i m e zero noted i n t h e f i g u r e w e r e due t o t r i m and f l i g h t d a t a b i a s and should be d i s r e g a r d e d throughout t h e run when comparing t h e s e d a t a . Except f o r t h e s l i g h t r o l l acceleration reversal noted i n the figure a t about 3 sec, the data f o r a t t i t u d e and a n g u l a r r a t e s show very c l o s e agreement. The h e l i c o p t e r a l t i tude d a t a agreed c l o s e l y between s i m u l a t i o n and f l i g h t and showed a d e c r e a s e of about 1.5 m d u r i n g t h e 5-sec p e r i o d .
F i g u r e 29 shows t h e d a t a f o r a run i n which t h e h e l i c o p t e r was f l y i n g s t r a i g h t and l e v e l a t approximately 38 k n o t s and t h e p i l o t made a l o n g i t u d i n a l c y c l i c s t e p i n p u t o f about 2 cm f o r s e v e r a l seconds followed by a p e d a l i n p u t of a b o u t 1 cm and a l a t e r a l c y c l i c i n p u t of approximately 2 cm. The same t r i m a i r s p e e d w a s used i n t h e s i m u l a t i o n , and t h e c o n t r o l p o s i t i o n changes made i n f l i g h t w e r e d u p l i c a t e d i n t h e s i m u l a t i o n as b e f o r e . The upper t h r e e p l o t s i n f i g u r e 29 show t h e t i m e h i s t o r i e s of t h e f l i g h t and s i m u l a t i o n c o n t r o l i n p u t s f o r t h e f i r s t 7 sec of t h i s run. The c o l l e c t i v e s t i c k w a s n o t moved d u r i n g t h e 7-sec p e r i o d shown. The t i m e h i s t o r i e s of h e l i c o p t e r a l t i t u d e and a i r s p e e d are given i n t h e f o u r t h and f i f t h p l o t s , where good agreement i s noted between f l i g h t and simulation. A s f o r the hover run, the differences i n Euler angles and body a n g u l a r rates a t t i m e z e r o should be d i s r e g a r d e d throughout t h e run when comparing t h e d a t a . Here t h e agreement i n p i t c h and yaw d a t a i s c o n s i d e r e d e x c e l l e n t , whereas t h e agreement i n r o l l d a t a i s o n l y f a i r . The remaining n i n e comparison c a s e s between f l i g h t d a t a and s i m u l a t i o n d a t a showed e x c e l l e n t agreement f o r t h e f i r s t few seconds i n each c a s e . I n some c a s e s t h e agreement remained good f o r s e v e r a l more seconds; i n o t h e r s t h e agreement degraded s l i g h t l y , as i n t h e examples shown i n f i g u r e s 28 and 29. I n g e n e r a l , t h e agreement between simulation d a t a and f l i g h t d a t a w a s believed t o i n d i c a t e t h a t t h e simulation mathematical model i s adequate f o r s t u d y i n g h e l i c o p t e r s l i n g load control systems.
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The h e l i c o p t e r w a s trimmed a t 0.1, 30, 60, and 90 k n o t s and t h e l i n e a r i z a t i o n algorithm described i n reference 19 w a s applied t o determine the l i n e a r system d e r i v a t i v e matrices A and B defined i n reference 19 and the system eigenvalues and eigenvectors. I n order t h a t t h e time-varying rotor inflow r a t i o s Vm and V t and t h e t a i l - r o t o r e f f e c t i v e p i t c h a n g l e OOt would n o t appear as variables i n t h e l i n e a r system, 30 i t e r a t i o n s w i t h t h e Vm, Vt,
and. eo, d i f f e r e n t i a l e q u a t i o n s (eqs. (26) and ( 4 7 ) ) w e r e performed a f t e r each
s t a t e v a r i a b l e p e r t u r b a t i o n w a s made so t h a t t h e s e t h r e e v a r i a b l e s would r e a c h semi-steady-state b e f o r e t h e l i n e a r system d e r i v a t i v e s were c a l c u l a t e d . The A and B l i n e a r system matrices are given i n table V I f o r t h e f o u r airspeeds.
Unpublished d e r i v a t i v e s obtained from Sikorsky A i r c r a f t , d e r i v a t i v e s c a l c u l a t e d by methods given i n r e f e r e n c e 7, and d e r i v a t i v e s given i n r e f e r e n c e 18 w e r e compared w i t h t h o s e i n t h e t a b l e and g e n e r a l l y good agreement w a s found.
The e i g e n v a l u e s and e i g e n v e c t o r s f o r t h e system A matrix w e r e c a l c u l a t e d and are a l s o given i n t a b l e V I . The upper h a l f - p l a n e of t h e l o c i of e i g e n v a l u e s a t the f o u r a i r s p e e d s i s shown i n f i g u r e 30. The e i g e n v e c t o r s were used t o i d e n t i f y t h e e i g e n v a l u e s w i t h r e s p e c t t o t h e h e l i c o p t e r modes of motion. The i d e n t i f i c a t i o n i s noted on t h e f i g u r e . The r e a l r o o t s r a n g i n g from -0.85 t o -1.0 are a s s o c i a t e d w i t h r o l l i n g v e l o c i t y and v e r t i c a l v e l o c i t y . The z’ero a t t h e o r i g i n corresponds t o t h e heading a n g l e $h. The root l o c i were compared with unpublished Sikorsky A i r c r a f t d a t a f o r a CH-53 h e l i c o p t e r , which i s s i m i l a r t o t h e CH-54, and w i t h d a t a a t 60 k n o t s g i v e n i n r e f e r e n c e 18. I n general, the l o c i trends w e r e found to be s i m i l a r b u t the agreement i n magnitudes was only fair.
The comparison o f t h e l i n e a r system m a t r i c e s and t h e e i g e n v a l u e s and e i g e n v e c t o r s o b t a i n e d from t h e n o n l i n e a r mathematical model w i t h d a t a from o t h e r s o u r c e s i n d i c a t e s t h a t t h e mathematical model does r e p r e s e n t t h e dynamics of a CH-54 h e l i c o p t e r f a i r l y w e l l .
The s l i n g l o a d p o r t i o n o f t h e mathematical model w a s v a l i d a t e d by compari n g measured frequencies of simulated pendulum, rocking, and bounce motions w i t h t h e o r e t i c a l v a l u e s and by p i l o t e v a l u a t i o n . The t h e o r e t i c a l and measured f r e q u e n c i e s o f t h e s e modes a r e g i v e n i n t a b l e V I I . I n each case t h e t h e o r e t i c a l values are based on the assumptions t h a t the helicopter attachment p o i n t i s f i x e d i n space and t h a t t h e cable i s i n e l a s t i c e x c e p t i n t h e bounce mode. I n each mode t h e measured f r e q u e n c i e s are s l i g h t l y h i g h e r t h a n t h e t h e o r e t i c a l values. This discrepancy i s believed t o be due t o the motion of the h e l i c o p t e r attachment p o i n t and t h e i n t e r a c t i o n between t h e bounce mode and t h e r o c k i n g modes. The agreement i s c o n s i d e r e d adequate.
As mentioned e a r l i e r , t h e suspension system c a b l e s p r i n g rate Ksc w a s c r i t i c a l i n t e r m s o f numerical s t a b i l i t y . T r i a l and e r r o r showed t h a t t h e s p r i n g r a t e had to be s e l e c t e d so t h a t t h e f r e q u e n c i e s a s s o c i a t e d w i t h t h e
c a b l e s t r e t c h would be no h i g h e r t h a n 2 Hz. A v a l u e of Ksc = 1.8 X l o 5 N/m
w a s used w i t h t h e 4536-kg l o a d , g i v i n g a t h e o r e t i c a l bounce n a t u r a l frequency of 1 Hz.
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P i l o t ' s Comments
The p r o j e c t t e s t p i l o t a c q u i r e d e x p e r i e n c e i n t h e Army CH-54 h e l i c o p t e r during the f l i g h t tests reported i n reference 17. This p i l o t also received additional f l i g h t training with long-line s l i n g loads i n connection with the p r e s e n t r e p o r t . Numerous s i m u l a t e d f l i g h t s b o t h with and w i t h o u t a s i m u l a t e d s l i n g l o a d a t t a c h e d were performed f o r t h e purpose of v a l i d a t i n g t h e s i m u l a t i o n mathematical model.
Several combinations of simulated c y c l i c s t i c k and pedal force g r a d i e n t s and dynamic c h a r a c t e r i s t i c s w e r e considered. The s e t used i n t h i s r e p o r t repres e n t a u s a b l e set f o r t h e s i m u l a t i o n . The VLDS and l o a d / l a n d i n g zone v i s u a l display system described earlier i s considered t o be marginally adequate f o r t h e s i m u l a t i o n of h e l i c o p t e r s l i n g l o a d o p e r a t i o n s . The l a c k of p e r i p h e r a l v i s u a l cues i n t h e h o r i z o n t a l and v e r t i c a l p l a n e s makes hover and t r a n s i t i o n s t o hover e s p e c i a l l y d i f f i c u l t . The computer-generated l o a d / l a n d i n g zone d i s -
. p l a y and t h e VLDS a l s o l a c k t h e d e p t h and n a t u r a l cues t h a t are so u s e f u l i n
a c t u a l f 1i g h t
The s i m u l a t e d h e l i c o p t e r a l o n e with and w i t h o u t winds and t u r b u l e n c e seemed l i k e a l a r g e h e l i c o p t e r i n a l l f l i g h t regimes. I t seemed underdamped i n yaw motions when approaching hover, and i t s v e r t i c a l motions seemed much more sens i t i v e t o c o l l e c t i v e movements t h a n t h o s e f o r t h e a c t u a l h e l i c o p t e r . The simul a t e d h e l i c o p t e r seemed slower than t h e a c t u a l h e l i c o p t e r t o respond t o c y c l i c step and pulse i n p u t s . Autorotation w a s simulated and seemed r e a l i s t i c t o touchdown.
The s i m u l a t i o n w i t h t h e e x t e r n a l l o a d seemed r e a l i s t i c i n s p i t e of t h e visual display shortcomings. Simulated f l i g h t s with the external load w e r e performed over t h e e n t i r e speed range and t h e ensuing load motions w e r e observed on t h e load/landing zone d i s p l a y . In a l l cases t h e load motions appeared r e a l i s t i c . The h e l i c o p t e r motions d e f i n i t e l y w e r e a f f e c t e d by l o a d motions, and hovering over a s p o t w a s very d i f f i c u l t , as it i s i n a c t u a l f l i g h t . A l l normal s l i n g load o p e r a t i o n s were t r i e d with and without winds and turbulence and were found t o be p o s s i b l e t o e x e c u t e b u t w i t h s l i g h t l y h i g h e r work l o a d t h a n i n a c t u a l f l i g h t . Simulator motion validation w a s not necessary since i t w a s thought t h a t f o r s t u d i e s o f s l i n g l o a d s t a b i l i z a t i o n and c o n t r o l system comp a r i s o n , simulator motion would n o t be required.
It is thought t h a t the simulation described i n the report can be used t o compare v a r i o u s systems f o r s t a b i l i z i n g h e l i c o p t e r s l i n g loads and improving helicopter sling load handling qualities.
CONCLUDING REMARKS
A generalized, full-flight envelope, real-time, piloted visual simulation of a single-rotor h e l i c o p t e r , suspension system, and e x t e r n a l load i s described and v a l i d a t e d f o r t h e f u l l f l i g h t envelope of t h e U.S. Army CH-54 h e l i c o p t e r
50

and c a r g o c o n t a i n e r as an example. The mathematical model d e s c r i b e d uses modif i e d nonlinear c l a s s i c a l r o t o r theory f o r both t h e main rotor and t a i l rotor, nonlinear fuselage aerodynamics, an elastic suspension system, nonlinear load aerodynamics, and a load-ground c o n t a c t model. The implementation of t h e mathematical model on a l a r g e d i g i t a l computing system i s d e s c r i b e d , and v a l i d a t i o n of t h e s i m u l a t i o n is d i s c u s s e d . The mathematical model i s v a l i d a t e d by (1) comparison of f l i g h t d a t a w i t h simulated d a t a ; ( 2 ) comparison of l i n e a r i z e d system c o e f f i c i e n t matrices, eigenvalues, and eigenvectors with c a l c u l a t e d values, manufacturer's d a t a , and d a t a obtained from f l i g h t tests; and (3) by p i l o t evaluation. A visual landing display system t h a t generates the p i l o t ' s forward-looking real-world d i s p l a y i s d i s c u s s e d , and a s p e c i a l head-up, downlooking load/landing zone d i s p l a y i s described. I t w a s the test p i l o t ' s opinion t h a t t h e s i m u l a t i o n d e s c r i b e d i n t h i s r e p o r t can be used t o compare various systems f o r s t a b i l i z i n g helicopter s l i n g loads and improving helicopter sling load handling qualities. Langley Research Center National Aeronautics and Space Administration
Hampton , VA 2 3665
October 23, 1978
51

REFERENCES
1. D a n i e l s , Glenn E . , e d . : T e r r e s t r i a l Environment ( C l i m a t i c ) C r i t e r i a Guidel i n e s f o r U s e i n Aerospace Vehicle Development, 1973 Revision. NASA TM X-64757, 1973.
2. Wilcock, T.; and Thorpe, Ann C.: F l i g h t S i m u l a t i o n of a Wessex H e l i c o p t e r -
A Validation Exercise. C.P. No. 1299, B r i t i s h A.R.C., 1974.
3, Seckel, Edward; and C u r t i s s , H. C., Jr.: 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 Rotors. Rep. No. 659, D e p . Aerosp. Mech. S c i . , P r i n c e t o n Univ., D e c . 1963.
4. B a i l e y , F. J., Jr.: A S i m p l i f i e d T h e o r e t i c a l Method o f Determining t h e C h a r a c t e r i s t i c s o f a L i f t i n g Rotor i n Forward F l i g h t . NACA Rep. 716, 1941.
5 . Amer, Kenneth B.; and Gustafson, F. B.: C h a r t s f o r E s t i m a t i o n o f L o n g i t u d i n a l - S t a b i l i t y Derivatives f o r a Helicopter Rotor i n Forward F l i g h t . NACA TN 2309, 1951.
6. Gessow, A l f r e d ; and Myers, Garry C . , Jr.: Aerodynamics of t h e H e l i c o p t e r . Macmillan Co., c.1952. (Republished 1967 by F r e d e r i c k Ungar Pub. Co.)
7. S e c k e l , Edward: S t a b i l i t y and Control of A i r p l a n e s and H e l i c o p t e r s . Academic P r e s s , I n c . , c.1964.
8. Amer, Kenneth B.: Theory of H e l i c o p t e r Damping i n P i t c h o r Roll and a Comparison With F l i g h t Measurements. NACA TN 2136, 1950.
- 9. Bailes, Edward E.; Diekmann, Vernon L.; W a t t s , Joseph C.; and Henderson, John C.: Instrument-Flight-Rules C a p a b i l i t y E v a l u a t i o n CH-54B (TARHE) H e l i c o p t e r . USAASTA Pro]. No. 71-01, U.S. Army, D e c . 1972. ( A v a i l a b l e from DDC as AD 908 656.)
10. Laub, Georgene H.; and Kodani, H i f u M.: Wind Tunnel I n v e s t i g a t i o n of Aerodynamic C h a r a c t e r i s t i c s of S c a l e Models of Three Rectangular Shaped Cargo C o n t a i n e r s . NASA TM X-62,169, 1972.
11. Liu, David T.: I n - F l i g h t S t a b i l i z a t i o n o f E x t e r n a l l y Slung H e l i c o p t e r Loads. USAAMRDL Tech. Rep. 73-5, U.S. Army, 1973.
1 2 . Wilson, John W.; and Steinmetz, George C. (With Appendix A by Roland L. Bowles) : Analysis o f Numerical I n t e g r a t i o n Techniques f o r R e a l - T i m e D i g i t a l F l i g h t Simulation. NASA TN D-4900, 1968.
13. P a r r i s h , R. V.; R o l l i n s , J. D.; and Martin, Dennis J., Jr.: Visual/Motion Simulation of CTOL F l a r e and Touchdown Comparing D a t a Obtained From Two Model Board Display Systems. AIAA Paper 76-1709, A p r . 1976.
52

14. Newman, W i l l i a m M. ; and S p r o u l l , Robert F. : P r i n c i p l e s o f I n t e r a c t i v e
, Computer Graphics. M c G r a w - H i l l Book Co. 1973, pp. 123-124.
15. Houck, Jacob A.; Gibson, L u c i l l e H.; and Steinmetz, George G , : A R e a l - T i m e D i g i t a l Computer Program f o r t h e Simulation of a Single-Rotor H e l i c o p t e r . NASA TM X-2872, 1974.
16. Fehlberg, Erwin: C l a s s i c a l F i f t h - , Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas With S t e p s i z e Control. NASA TR R-287, 1968,
17. D i C a r l o , Daniel J.; Kelley, Henry L.; and Yenni, Kenneth R.: An E x p l o r a t o r y F l i g h t I n v e s t i g a t i o n of Helicopter Sling-Load Placements Using a ClosedC i r c u i t T e l e v i s i o n as a P i l o t Aid. NASA TN D-7776, 1974.
18. Tomaine, Robert L.: F l i g h t D a t a I d e n t i f i c a t i o n of S i x Degree-of-Freedom S t a b i l i t y and C o n t r o l D e r i v a t i v e s of a Large "Crane" Type H e l i c o p t e r . NASA TM X-73958, 1976.
19. Dieudonne, J a m e s E. : D e s c r i p t i o n of a Computer Program and Numerical Techn i q u e f o r Developing L i n e a r P e r t u r b a t i o n Models From Nonlinear Systems Simulation. NASA TM-78710, 1978.
53

TABLE I.- VALUES OF PARAMETERS FOR CH-54 HELICOPTER

[Subscripts m and t denote main and t a i l rotor]

Xmr,hr m
ymlh1m

............................... ...............................

-0.33 0

Zmrlhr m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -2.26

Xtr,hr m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -13.74

ytrlh1m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.84 ztrlh,rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -2.22

Xwt,hr m yWtlhlm

............................... ...............................

-0.51 0

zwt,h1m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.37

Xp s , h l m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.77

ypslh1m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-0.53

Zp s , h l m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.11

F S C G , m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.20

WLCG,m...............

. . . . . . . . . . . . . . . . . 4.28

BLCG,m... ............................. 0

x a l h l m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.33

yalh,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

zalh,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.24

Ixxlh1kg-m 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 800

Iyy,h,kg-m 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.04X1o5

kg-m 2 'z.z1h1 I x z l h lkg-m 2

........................... .............................

1.78 x lo5
11 400

em,m.................................

0.610

Ibmlkg-m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.75 X lo3

Glkg-m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.0

&,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.97

~ ~ ~ , .s .e .c. . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.20

63m,rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

elm, rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.183

Om . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.115

54

TABLE I.- Continued

'sm r rad $)smrrad

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-0.0524 ............................... 0

R,. rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.5

Ggov. N-m/(rad/sec) . . . . . . . . . . . . . . . . . . . . . . . . . . 85 160

Kdgov. N-m/(rad/sec) . . . . . . . . . . . . . . . . . . . . . . .

1.32 x 105

&. N-m/(rad/sec) . . . . . . . . . . . . . . . . . . . . . . . . . 1.572 X 106

Kgovr N-m/(rad/sec)
. . . IP t ' kg-m2 . . . Imr. kg-m2

.... ..... ......

. . .

. . .

. . .

...... ...... .......

. . .

. . .

. . .

...... ...... .......

. . .

. .

. .

. .

833.3 4325
3 1 310

Teng. sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.50

GBe. rad/rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.281

GA+. rad/rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.133

GBxlon. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.363

GAxlat. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.475

Gech. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.00037

G rad/(rad/sec) . . . . . . . . . . . . . . . . . . . . . . . . . .

0. 727

Bq' GAP. r a d / ( r a d / s e c )

. . . . . . . . . . . . . . . . . . . . . . . . . . -0.096

Getr. rad/(rad/sec) . . . . . . . . . . . . . . . . . . . . . . . . . .

0.335

Get@' rad/rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.133

KcO. rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.128

Kcl. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.955

Kc2. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.361

KC3. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.096

K,.,rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.824

Kc5. rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0494

Kc6. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.64

Kc7. rad/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.09 Kfe. N-m/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0243

KSc. N/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8X105

mhrkg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610

at. per rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.73

55

TABLE I.- Concluded

B t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.92

bt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

c t . m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.343

et.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.127

-Ibtl
\t.k

kg-m2 g-m

.

. .

. .

..... ......

. .

. .

. .

. .

. .

. .

. .

..... ......

. .

. .

. .

. .

. .

.... ....

.

.

13.88 11.84

T ~ s~ec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.20

T A ~ ~se,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.20

elt1rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -0.140

s l m .................................

2.44

CTt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.179

est. rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

(Pstlrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.57

63t. rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.78

Bm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.97

,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

cm.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.661

aml per rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.73

ua1 rad/sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.0

pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0

ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

ekf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.5

it- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

56

. TABLE I1 .VALUES O F PARAMETERS FOR 2.4-m BY 2.4-m BY 6.1-m

CARGO CONTAINER

Ixxg. kg.m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124

IyV%. kg.m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 610

I z z ~ . kgm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 610

I x z g , k g-m 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

mR. kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4536

Xa.R. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

yaIR. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

Za.R.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -6.1

x ~ ~ ,rn ~ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.05

Y , ~ . ~ .rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1.22

Zcl.R. rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22

x ~ ~ .m~ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.05

yC2.%. rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22

zc2.$. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22

x ~ ~ m. ~. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -3.05

yc3.g. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22

2C3'R' m
xc4.%. m

............................... ...............................

1.22 -3.05

yc4.g. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1.22

Zc4. G' m Vo. rn/sec

............................. ...............................

.

.

1.22 0.305

Vf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.25

%. N/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 000

G. N/(m/sec) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7100

RcO. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30.5

57

TABLE 111.- REQUIRED COMPUTER TIMES

calculation
Helicopter only Cockpit interface Winds and g u s t s Visual landing
display system Load/suspension
system Load l a n d i n g zone
visual display T o t a l simulation

- Computer t i m e , m s e c , f o r mode

Reset

Hold

Operate

2.5

1.7

2.0

.3

.2

.3

.o

.1

.2

.7

.7

.8

.9

09

1.0

1.5

1.4

1.9

5.9

5 .O

6.2

TABLE 1V.- DYNAMIC CHARACTERISTICS OF VLDS

Drive
A 1t i t u d e Lateral Longitudinal Roll Pitch Yaw

Frequency, rad/sec 27.9 24.1 22.8 18.6 24.3 22.6

Damping ratio

0.64

.6

I

.3

.7

.7

.7

58

TABLE V.- TRIMMED VALUES OF MAJOR VARIABLES AT FOUR AIRSPEEDS [Subscripts m and t denote main and t a i l r o t o 4

Variable

V a l u e a t airspeed. knots. of .

0.1

30

60

90

. . . . . . Xlon ' c m
. . . . . . xlat. c m

-5.48 -0.12

-3.98 -1.09

-2.55 -1.79

-1.10
-2.79

2.04

. . . . . . XPed' cm * = *
xcol. c m

16.4

. . . . . . AiC. deg

-0.95

. . . . . . BiC. deg

-4.27

eOm. deg . . . . . . 16.3

Oat, deg . . . . . . 15.2

. . . . Uas.h. m / s e c

0.05

. . . . m / s e c

0

. . . . was.h. m / s e c

-0.001

. . . . . . @h. deg

-2.8

. . . . 0h. deg

-1.3

. . . . . . af. deg

-29.9

Bf. deg . . . . . . 0

. . . . m / s e c

0.05

. . . . . alsm. deg

4.3

. . . . . blsm. deg

-0.95

. . . . . aOsm. deg

5.82

. . . . . alst. deg

0.004

. . . . . blst. deg

0.001

. . . . . aOst. deg

2.14

. . . . . . 'mr. h. N . . . . . . Ym,.h. N

-2947 -2204

. . . . . . 'mr. ht N
. . . . . 'mu. h. N-m

-1.33 x 105 -1.93 X 104

1.05 13.7 -1.27 -3.10 14.9 12.0 15.4 0 -0.45 -2.2 -1.7 -15.0 0 15.4 4.1 -0.73 5.59 0.83 0.18 1.57 -2605 -1831 -1.34 x 105 -1.53 x 104

0.06 11.9 -1.50 -1.99 13.8 9.3 30.9 0 -1.46 -1.6 -2.7 -6.9 0 30.9 3.8 -0.49 5.31 0.96 0.26 1.09 -1828 -1796 -1.34 x 105 -1.21 x 104

-0.48 13.1 -2.03 -0.86 14.5 8.8 46.2 0 -4.06 -1.7 -5.0 -6.9 0 46.3 3.7 -0.53 5.35 1.33 0.44 1.14 -1635 -1955 -1.34 x 105
-1.33 X 104

59

TABLE V.- Continued

Variable

- V a l u e a t airspeed, knots, of

0.1

30

60

90

. . . . . Mmr,hr N-m

-939

- - Nmr,hi N-m

e

1.20 x lo5

-3420
9.44 X l o 4

-7734
7.54 X l o 4

-8927
8.37 X l o 4

. - X t r , h , N

.

-0.62

. Y t r , h l N

. 8699

Ztr,h, N

. . 0.17

. . . . . L ~ ~ N-,m ~ ,

1.93 x lo4

. . . M t r , h , N-m

-2288

. . . . . Ntrlh, N-m

-1.20 x lo5

. . . X f , h , N

-0.010

. Y f , h # N

. O

. . . . 'f ,h' N

. . -0.018

-99.6 6868 25.3 1.53 x 104 -721
-9.4 X l o 4
-1161 0 1046

-91.4 5414 19.4 1.21 x 104 -254
-7.4 X l o 4
-4401 0 376.5

-138.4 5912 29.5 1.33 x 104 59.2
-8.09 X l o 4
-9904 0 865.3

Lf ,h' N-m

0

. . . Mf ,hr N-m

. . 3227

. . . . . N f , h , N-m

-0.003

. hhr m . . . . .. 30.5

. . . . . . P, kg/m3

1.23

s l N/m2 . . . . . . 0.0016

. . . . . Qml r p m

.

184.5

Cm

. - . 0.00640

Pm . . . . . . . . . 2.42 X

. . . Vm . . . . . . 0.0566

. - - Am

0

. 0

-0.057

. . . . . . earn,N-m

1.19 x lo5

0 4142 -340.1 30.5 1.23 145.6 184.5 0.00646 0.0726 0.0388 -0.041 9.43 x 104

0 7988 -1252 30.5 1.23 582.3 184.5 0.00642 0.145 0.0217 -0.031 7.51 x 104

0 886 8 -2824 30.5 1.23 1310 184.5 0.00643 0.216 0.0146 -0.042 8.33 x 104

60

TABLE V. Concluded

Value a t airspeed. knots. of .

Variable

.0.1
.,T N . . . . . . . 1.33 lo5

H,. N . . . . . . . 89.9

.,J N . . . . . . . 5.04

. . . . . . Qt. r p m

835.6

.30
1.34 lo5
2381 1135 835.6

.60
1.33 4190 1683 835.6

.90
1-33 105
6628 2775 835.6

61

TABLE VI.- HELICOPTER LINEAR SYSTEM MATRICES, EIGENVALUES, AND EIGENVECTORS

( a ) V a l u e s a t 0 . 1 knot

TRINMED V E L O C I T Y =

-1 KNOTS

7 POINT F O P N U L A

30 ITERATIONS

INDEPENDEW VARIABLE TOTAL INCRENENTS

m3048E-01 -3048E-01
.304U€-Ol -1745E-02 -1745E-02 -1745E-02 -1745E-02 ~ 1 754E-02
*1745€-02

.i745c-n2 1745E-02
.1745E-C2 1 7 4 5F-02

ucg, h
-.1396E-01 s7121E-02
-.OZO+E-OZ .1361C-O1 .527ZE-02 .1061E-02
0. 0. 0.

"cg.h
-.73271-02 -.2995E-01 -.5904~-02 -.3444E-01
e2554E-02 +1585E-G1 0. C. 0.

A mmix

"c8.h
-.83046-02 -.5138E-C>2 -.3337~+00 -.3684€-02 -.6207E-02
m8314E-02 C. 0. C.

h '
-116471-01 .1519E+00
-.i7n8~-oi -.7563%+00
.1254E*00 -.ZZG9E-01
.1000E+01 0. 0.

qh
.2527E+00 -.7543i+OO -.3789~-01 -.126ZE+O1 -.2170i+00 -.6732€-01
.1377€-02 .9988€+00 -.4872%-01

r
m1701E-02 e1966E400 -.1012E-01 -7811E-01 -.5994€-02 -.2458E+00 -.2209E-01 .4571E-01 .9991E+00

h '
-1824E-09
.9792 E +o1
.4775E+00 -e53671106
.3961E-07 -1178E-05 0. 0. 0.

h '
-.9804€+01 -.1056E-01
.2165E+00 -.50826-20 0. -.1+91E-18 0. 0. 0.

*h
-.lZ7€-18 .1487E-18
-.7667€-17 -.50828-20 0. -.1491t-l8 0. U.
0.

8 HATRIX

.9754i+Ol .2823€-03
--...5282E*00 .1965E-03 3857f+01
12451-03 0. 0.
0.

..
~7238E-13 .9767E+01 -.7667E-17 .2015E+Ot
-.8009E-l0 1191E+Ol
c.
0.
0.

Born
-.2019E+Gl -.1508E+01 -.9098E+OL -.7471E+00 -.1554E+01
.7@57€+01
0. 0.
0.

8ct -.54(r9E-03
4165 E + O l 1 378E-03
-..1941E+01
-.9576E-01 42 bOi+Ol
0.
0.
0.

P E A L PARTS O F EIGENVALIJES

ROLL@.)

SHORT PERIOD(SP)

-.8493EE*JO

-.63450E+OO -.33748E+CC

I R A G PARTS OF LIGENVALUES

0.

0.

0.

EIGENVECTORS
%.h

"cg,h

"cg. h

R .3003tE+00

.9430RE+Ot

-81523E-01

S P I F a L (S) -.ZC704E+OC 0.
'h -68313E-01

HEADING (HI -.22678E-17 0.
h ' -.19306E-01

PHUGOID(P)

.lO13OC+I)O

.lGl30€+OC

h ' -.26074F-01

h ' -+5108bE-01

DUTCH ROLL(DR)

.114511+00

.11451t+00

h ' -24199E-01

*h ~29564E-01

'-.58@93t+!lU
sp J
\-.69560E+30

-.79722€+0C -.12582E*00

-.86512E-01

-+31491E-01

-.6473@Et00

.11696E-G2

.21029E-01 -30674E-02

.378228-01 .91051E-01

.5091ZE-01 .24837E-02

-.3600bE-01 -.222196-01

-.57939E-01 -.26910€+00

.17156E+OO -.12123€-01

-.57536t-02

-.82955€+00

-39610E-19

.10000E+Ol

62

TABLE VI.- Continued

&g,h
ecg ,h
*cg,h
Ph
t9h

(b) Values at 30 k n o t s

- TRIMMFO VELOCITY

30.0 KNOTS

7 POINT FOhtJULA

30 ITERATlONS

INOEPENOENT VARIABLE

__ _-

-._>g,h

-3048E-01

-..wcg,h - .3OIB€-Ol

--.Bcg,h . .3048€-01

=-h-.1745E-02

h '

117456-02

ph

.1745E-O2

h

-1745E-02

.1?45€-02

_h

-17456-02

TlTAL

INCREMENTC

%,h
-.ZRO~E-~I .1Z65E-Ol
-.2238€+00 -1642E-01 -172OE-02
-.9055E-OP 0. 0.
0.

'c&h
-.6359~-0~ -.6052€-01 -.1553E-01 -.1596E-01
-5378E-03 .2371E-01 0. 0. 0.

A MbTRIX

-. "cg,h 1Z Z I E-C 1
-.9035€-02 -.5558E+00 -.97016-03 -.8478€-02
-5737E-02
0. 0.
0.

h '
-.-.1199E-01
-.4713E+00 33341-01
-.BBO8E+OO .12ZZE+OO
-.74251-02 .1000E+Ol
0. 0.

h '
.8423E+00 -.84638+00
.1580t+02 -.1324€+01 -.Z409E+00 -.+336€-01
.1095€-02 .9993€+00 -137796-01

h '
U 2 +OE-03 -.1508E+OZ -.SCO9€-02
.1562E+OO -.2646E-02 -.3949E+00 -.2897€-01
,37778-01 .9997E+OO

'h
- 2 675 E-0 7 .9795€+01 .3702E+00 -t1655E-06 -1255E-07 -3567E-06 0. 0. 01

-. *h 9802 E +01
-.1073E-01 .2840E+00 a3970t-21
-.97ClE-20 -.3812E-20 0. 0. 0.

*h
908 BE-19 .187Zt-ZO rZ324E-20 .3970€-21 -.97*1€-2(1 -.3812€-20 0. 0. 0.

B;C .1005E+02 .1351E+00 .83646+01
.1314E-01 -.3750E+01 -.88491-01 0. 0. 0.

4c
.6138E-O7 r9050E+01 .309kE-05
.2021 E+OZ ~37 4 1E-07 .1193E+01
0.
0. 0.

.1162E+Ol -.9385E+QO
.6510E+Ol
0. 0. 0.

B MATRIX
-.1902€-01 -.3890€+01 0. 0. 0.

REAL PARTS O F EIGENWALUFS

ROLLCR)

SPIRAL(S)

-.89431E+00

-.51224E-01

I M A G PARTS OF EIGENVhLUES

0.

0.

EIGEUVECTORS

Ucg,h

"cg,h

R -.16631E+00

-.66433E+00

HWING(H) .1127OE-lR
0.

PHUGOIDD(P)

e62776E-01

-627768-61

DUTCH ROLL(DR) -r12986E+00 -.12986E+00

.28606€+00 -.28606E+00

.66800E+00 -.66800f+OO

SHORT PERIOD(SP1

-.54060€+00

-.54060E+00

wcg,h -.71209E+OO

h ' -.95506E-01

h ' e97492E-02

I .36125F-01

h ' .10795€+0O

Oh -.12419E-01

*h -.399718-01

S .71857€-01 -.57018E+00

H e1290PE-17

~26637E-17 -rSPSC7€-18

.32664€-20

-.425k9E-25

.11257E-10

.1871SE-18

.42116€-20

*10000E+01

.98380€-01

-3969kE-02 -.69572E-02

-.76027E-02

-.63714E-02

-.113C7€-01

-.56697E-02

.2281?€-01

.24407€-01

.17297€-02

-.11951€-01

-4979CL-01 -.18702E-02

.Zl2791-02

-.247hlE-01

e23334E-01

.28407E+CO -.24127€-01

e17039E-01

.108985-01

e39045E-01

-.69323E+00

.1284ZE-01

.14830€-UE

.21678C-01

.78622€-02

63

TABLE VI.- Continued

(c) Values a t 60 k n o t s

- T R I M M E D VtLLlCITY

60.0 KNOTS

7 POINT FOPIIULA

30 ITERATIONS

INOEPENOENT YAPIA8LE TOTAL INCRCqENTS

~304EE-Gl a304EE-01
~3048E-01 -1745E-02 a1745E-02
.1745E-02 41745E-02 e1745E-02 .IW~E-OZ

5~
AIC
eom
t

.1745E-02 .1745€-02 .1745E-02 t 1 7 4 5 f -02

%,h "cg,h *cg,h Fh sh fh

A MATRIX

Ucg,h -.3330€-01
.6410€-02 -.1251€+00
.1269E-01 -3674E-02 -.5580E-02 0.
0. 0.

"eg,h -.47928-02 -.9340t-01 -.2696E-01
m4809E-03 -.1469E-02
-3119E-01
0.
0. 0.

"c8.h -.1466E-C.1 -.1445€-01 -.7800E+OO
,50336-02 -.1098E-01
nlE71E-OZ 0.
(1.
0.

h ' -.8590E-OZ -r1637E+01 -.5264€-01 -.9786E+00
.1196E+00 .lZEBE-Ol *1000E+01
0.
0.

h ' .1939E+01 -.8464E+00 .3092E+02 -.1317E+01 -.2723E+00 -.5417E-Ol e1289E-02 .9996€+00 -.2719E-01

h ' .2016€-02 -.3037E+OZ -.1117€-02 .2433E*OO .1300E-03 -.5570E+GO -.4743F-01 .2716€-01 .1001E*01

h ' ,27141-07 .9792E+Ol .2661E+00 .llOZE-06 .1770E-08 -.2561E-06 0. 0. 0.

h ' -.9795E+01 -.lZbZE-Ol
.4646E+00 -.741ZE-21 -.1959E-20
.1059€-19 0. 0.
0.

.h' 1.33561-20 -.9036€-21 -.Z582E-19 -.7+1ZE-Zl -.1959€-2U
.1059E-19 0. 0. 0.

e.h cg,h
*cg,h ph ?h h ' h eh *h

E MATRIX

BIC
.103@Ei02 .4378E+00 .Z248E+02 -.1322€+00 -.3564E+U1 .1022E-01 0. 0. 0.

4C
e5083E-05 .9800E+01 ,1866E-03 .2017E+02 .2530€-05 .1192€+01 0. 0. I..

eom
-.4717E+01 -.1391E+00 -.lU34€+03
.2833E+Ol -.6393E+GO
.5376€+01 0.
0. 0.

ect
-.1668E+00 .44781+01 e3579E-01 -2117E101 .2769E-01
-.4560E+01 0. 0. 0.

RF.AL PARTS O F EIGENVALUES

ROLL00

HEADING(H)

-.95639€+00

-.68986E-20

IPAG PARTS OF EIGENVALUES

0.

0.

. EIGENVECTORS "cg h

"c%,h

R -.74926E-01

-.3342RE+00

SPIW(S) e56569E-01
0.

PHUGOID(P)

.27343E-01

a27343E-01

SHORT PERIOD(SP)

-.h0458€+00

-.60458€+00

.27284E+00 -.27284E+00

.57861E+00 -.57861E+00

DUTCH ROLL(DR)

-.33015E+OO

-a33015€+00

a98819Et00 -.96819E+00

"cg,h -.925\)9E+tO

h ' -.10735E+OU

h ' .35915E-02

h ' .3333@€-01

h ' .11390E+00

h ' -.47007E-O2

h ' -.34782€-01

64

TABLE VI.- Concluded

(d) V a l u e s a t 90 k n o t s

. TRlHflEO V E L O C I T Y

90.0 KNOTS

7 POINT FORMULA

30 ITERATIONS

INDEPENOENT VARIABLE TOTAL 1NCREMENTS

.3048€-01
30rBE-01 .304BE-01 a 1 7 45E-U2 .174SE-(rZ .1745E-O2 m1745E-02 -1745E-02 .1745€-(12

m1745E-02 .1745f-02 47456-02 .17458-02

A MATRXX

Ucg.h
-.-.4276E-01 -5430E-02 7655i-0 1

-. "cg.h
-.5663E-02
1 2 3 3 E +00 -.4154t-01

. i i s a ~ - i ~ i .15*6--Ul

.3325--72

-.3643 .--de

-.4 397- -02

.3452f-<l

0.

n.

( . c.

0.

0.

wcg,h
-.17298-01 -.2Zl2€-Ol -.9147E+00
,9965E-L2 -rll79E-C1
., 5 3 2 ~ E - 0 2
C.
#I
0.

h '
-.9421E-02 --.4238E+01 -.&078€-01 -.9b89;t35
.1174E+00 .25Ci5€-01 .10GGE+01
U. 0.

'h
,447Btt01 -.7714EtOO
.465bc+02 -.iZ45zt11
-.2@24E+00 -.5239E-O?
rZ619E-02 .99Jbi+OO -.2992€-01

'h
-.lkO4#-06 -.4560E+Ot
.16't4€-02 .307OE+bO .735bE-@? -.6505F+00 -.q785E-O1
.%Q80'-01 .1903E+01

h '
.3563E-O7 .9764€+01 .Z'?ilE+OO ,293rl-Vb
-.4dZt.E-.18 -.t67OE-O6
0. 0. 0.

*h -.929kE-20 -*1859E-19
,165tE-19 .296>t-20 -.b353€-21 -.la63t-i~
b. 3.
V.

BIc
.1099Et02 .1002f+01 .362Bt*r)Z
-.-.4119~+oo
-.355bE+01 9386Z-L1
I..
L. 0.

4,
.2015f.-J4 .9810Et01 ,6357f-03 .201a~t02 .53?6E-05 s1177tt01
C.
0.
0.

B IIATRIX

em
-.6631E+01 -.2125E+OO -.1196E+03
.. 4 5 5 5 ~ + 0 1
.1023E+OU S798EtCl
0.
(.
0.

8Ct
-.2725E+OO .5161i+01 -5712i-01
-.. t 4 6 z ~ + o l e7364i-01 5249c+Ji 0. c. 0.

ROLL(R)

HEADING(H)

l E A L P A R T S 3F EIGENYALUFS

-.95300Et00

-.,?9966E-19

I M A 6 PARTS OF EIOENVALUES

0.

0.

EIGENVECTORS Ucg,h

R .Z9092€-01

.k9194E-01

H .37692€-18

,331638-19

S r21475E100 -.83181EtOC

SPIRAL(S) ,89335841

PHUGOID(P)

.10409€-01

.lGk09E-01

SHORT PERIOD(SP)

-.6+693€+00

-.64693€+00

DUTCH ROLL(DR) -.42291E+00 -.42291ttOO

0.

wcg.h .98748Et00

'h -95330E-01

h ' ,72726E-04

r -.30462E-01

h ' -.10284€+00

h ' -87635E-03

'h .32075E-01

-.16052€-19

-.26148E-Z0

.08667€-21 -.29738€-19

-r13788t-l@ -.21832t-20

.10000E+01

.27103E-02

-.23195E-01

.10484€-02 -.40927€-01

-.Z1936€*00

-.19235€-02

-.46001E+00

65

.- TABLE V I 1 THEORETICAL AND MEASURED FREQUENCIES
OF LOAD MOTIONS

Mode
Vertical bounce Longitudinal pendulum L a t e r a l pendulum Longitudinal rocking Lateral rocking

Frequency

Theoretical, Hz

Measured,
KZ

1.0

1.2

.083

.097

-083

.091

.58

* 74

1.1

1.4

66

k naJ
67

$4
a,
U
a
0 v
ad
d
ca,
$4
u 0
68

Manual and automatic c o n t r o l system equations (12) to (16)
(c) Control system model. F i g u r e 1.- Continued.
69

W
cg,h

U gust,h
).
V gust,h
I_.___)
W gust,h *

Uwind,h +
Vwind,h *
w
wind,h *

Rotor equations (17) to ( 4 6 ) with

main-rotor parameters

- xr,hm
r' ,hm
_____)
- r' ,hm
______)
r' ,hm
M r,hm
____)
- Nr,hm
P
U as,h

V as,h
_____)
A Was h

'Tm
_____)
Tm
_.___)
x
m
_____)
m '
_____)

( d ) Main-rotor (subscript m) m o d e l . Figure 1.- Continued.

70

"c5z.h

Ugust,h

V gust,h
c
W
gust,h
- U wind, h
- V
- wind,h

Rotor equations
(17) to ( 4 7 )
with tail-rotor parameters

'I-. ht
-
'r.ht L r ,ht

- _W_wi_n_d,_h _, h '

3r,ht
_____)
Nr,ht

h ' r
h

(e) Tail-rotor (subscript t) model. F i g u r e 1.- Continued.
71

(f) Engine dynamics and governor model.
- Figure 1. Continued.
72

U
as.h

u was h

Fuselage

aerodynamics

.-
Ah
_______+

equations (49) to (58), (86) t o (89);

figures

(5) t o (11)

(9) Fuselage aerodynamics model.
- Figure 1. Continued.

73

Ug u s t , &

Load

xR

___.____)

aerodynamics

V
gust,%

equations

W
gust,R

(90) t o (96)

-. Uwind, R
_______)
Vwind, R
Wwind R PR

N"

c

( h ) Load aerodynamics model. F i g u r e 1.- Continued.

74

Loadground contact equations
to (66)

__I___)
ZC,R
MC.R NC,R

( i )Load-ground c o n t a c t model. F i g u r e 1.- Continued,

75

Load suspension
system equations (67) t o (77)
-
‘h

M t,h
N t,h

‘t,Q

(j) Load suspension model.
- Figure 1. Continued.

76

zLh CMh CNh

Equations of motion with helicopter parameters (78) t o (85)

fk) Helicopter e q u a t i o n s of motion. F i g u r e 1.- Continued.

77

CFx.Q
CFY, Q CF
z,Q
______+
C=R

Equations of motion with load parameters (78) to (85)

( 2 ) Load e q u a t i o n s of motion. Figure 1.- Concluded.

78

k
al
79

(a) Body axes.
X W

(b) Shaft axes.

(c) Control axes.

(d) Flapping angles.

F i g u r e 3.- H e l i c o p t e r body axes, s h a f t axes, c o n t r o l axes, and flapping aogle definitions.

80

\Helicopter cable attachment point
(*haye’’ha, e’ =ha, e I
Z e (Down) Figure 4.- Load suspension cable a n g l e d e f i n i t i o n s .
81

7.5
s .o
2.5
D
-2.5
-5 .o
-7.5 -10.0
Figure 5.- Fuselage incremental lift as a function of a n g l e o f attack.
82

I

-2

I x 10'

F i g u r e 6.- Fuselage i n c r e m e n t a l l i f t as a f u n c t i o n of s i d e s l i p angle.

83

7-

~-I N 6

-8

i”
e L -12

-16

-1

-9

x 10‘

Figure 7.- Fuselage s i d e f o r c e as a f u n c t i o n of s i d e s l i p angle.

84

I x LO’ F i g u r e 8,- Fuselage r o l l i n g moment as a f u n c t i o n of s i d e s l i p angle.
85

m E

-6

I

!I-

-24

-30

-32

-24

A -8

8

16

24

Figure 9.- Fuselage incremental p i t c h i n g moment as a f u n c t i o n of angle of a t t a c k f o r d i f f e r e n t v a l u e s of incidence a t t h e t a i l .

86

F i g u r e 10.- Fuselage incremental p i t c h i n g moment as a function of s i d e s l i p angle.
87

h

L

0

1 x 10'

Figure 11.- Fuselage yawing moment as a f u n c t i o n o f s i d e s l i p a n g l e f o r d i f f e r e n t v a l u e s of a n g l e of attack,

88

89

trl
In
0 rl I
a,
u) I
4
4
I a,
[o
0 4 V a,
rl
0
;
0
V
rl
0 k 4J
d
8
90

RESET

IDLE

RELEASE ERASE

PRINT RELEASE

( a ) Mode c o n t r o l switches.

I DECIMAL (-) 1 POINT

(b) Data e n t r y keyboard.

(Address f i e l d )

(Magnitude f i e l d )

(Exponent f i e l d )

( e ) D i g i t a l decimal display. Figure 14.- Mode c o n t r o l , d a t a e n t r y , and d i g i t a l d i s p l a y
systems on s i m u l a t i o n control console.
91

92

93

94

95

Figure 19.- Simulator c o n t r o l system analog computer (on the l e f t ) .
96