Independence of helicopter rotor derivates under non-uniformity of induced velocity distribution at low forward speed

TECHNISCHE HC

VLIEGTUIGBOL Kanaalstraat 10 A.. ...

_ CpA Report

. 3 SEP. 1957

Kluyverweg 1 - S DfcLFT

THE COLLEGE OF AERONAUTICS

CRANFIELD

INDEPENDENCE OF HELICOPTER ROTOR DERIVATIVES

UNDER NON-UNIFORMITY OF INDUCED VELOCITY

DISTRIBUTION AT LOW FORWARD SPEED

by

REPORT NO. 110 ITOVKvESSR. 1956 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D Independence of h e l i c o p t e r r o t c r d e r i v a t i v e s u n d e r non-unifon-.iity of i n d u c e d v e l o c i t y d i s t r i b u t i o n

a t lov/ forward speed b y

-¥,W. B r y c e , B , A . S c , ( T o r . ) , D.C.Ae. Smi/I/iRT

A r a d i a l p a r a b o l i c i n d u c e d v e l o c i t y d i s t r i b u t i o n a g r e e i n g c l o s e l y \ 7 i t h f l i g h t measurements h a s b e e n u s e d f o r t h e h o v e r i n g c a s e . To t h i s h a s b e e n added a second induced v e l o c i t y d i s t r i b u t i o n , v a r y i n g l i n e a r l y from t h e f r o n t t o tlie r e a r of t h e r o t o r d i s c , t o a l l o w f o r t h e e f f e c t of forward s p e e d . The magnitude of t h i s second i n d u c e d v e l o c i t y term depends on t h e advance r a t i o yL,

V a l u e s of t h e f o r c e c o e f f i c i e n t s C„ and 0.^-*

n It)

the flapping coefficients a , a. and b., and the rotor derivatives x , z , y.„, x , z , x , z and y have been

q* q* "T* u' u' w' w ''v

calculated for a typical case for the low forward speed region (n = 0 - 0,14) for both liniform and non-uniform induced vel-ocity and the results compared. Additional values of the flapping coefficients have been calculated for the speed range |i = 0,14 - 0,24 and the results compared va.th flight measurements and vn.th values based on the Msjigler induced velocity distribution. Good agreement has been obtained,

The values obtained for the rotor derivatives show that the effect of non-uniforr-i induced velocity is almost negligible except in the case of z which is a very small derivative,

-2'

CONTËINTS

last of Symbols 3 1 • Introduction 6 2, Notation 6 5» The Flov7 Relative to the Rotor Disc 7

4, The PIOT^ Relative to a Blade Element 8

5, The Induced Velocity 9 6, The Thrust Coefficient 15 7, The Feathering Coefficients 15 8, The H Force Coefficient 17 9, The Side Force Coefficient C^g 18

10, The Rotor Stability Derivatives 19 11, The Force-Angular Velocity Derivatives 20

X , y„ and z

12, The Force-Velocity Derivatives x and z 22 13, The Force-Velocity Derivatives x and z 23

14» The Force-Velocity Derivative y 25 15. Calculation of Force Coefficients, Flapping 26

Coefficients and Rotor Derivatives for a TyiDical Case lé, Discussicai 27 17* Conclusions 30 References 30 A]7pendices 32 Figures 41

-3-IJST OF SnZBOI£

a Blade section lift

curve

slorjc

a Blade coning angle

a. First harmonic longitudinal flapping coefficient

A Parameter in expression for X_ (S-lé)

I

_{Parameter in ejcpression for Xy (5~léA)}

A Blade collective pitch angle

Aj Coefficient of -cos ^ in expression for cyclic

feathering

b Number of blades

b. First harmonic lateral flapping coefficient

B Parameter in expression for

"k^

(5-16)

B' Parameter in expression for ?w, (5-1 éA)

B. Coefficient of -sin

^

in expression for cyclic

feathering

c Blade chord

0 /iTbitrary constant

Cj. Blade section lift coefficient

Ojj H farce coefficient = H /

%R^p(n

R ) ^

C J Thrust coefficient = T / 7cR^p(fi R ) ^

Cyg Lateral force coefficient =

Y^/viR^pi^ R ) ^

D Drag force on blade

F Aerodynamic force on blade

H Drag force in plane of rotor disc

i Incidence of rotor disc

Ij Blade moment of inertia about flapping hinge

L Lift force on blade

M. Moment of aerodynamic forces about flapping hinge

-4-List of Symbols (Contd,)

M_ iioment of dynamic forces about flapping hinge

P Rate of roll (positive OY "* OZ)

q Rate of pitch (positive OZ •• OX)

r Radial distance along blade from hub

H Blade radius

T Rotor thrust force

u,v,w Disturbance velocities along OX, OY, OZ respectively

U Resultant air velocity relative to blade element

(~U^)

Up Air velocity component perpendicular to blade and

to the rotor cone

Urn Air velocity component perpendicular to blade and

tangentiel to the rotor cone

V Velocity of forv/ard flight

V' Resultant air velocity relative to rotor disc (5-10

X,Y,Z Forces along OX, OY, OZ respectively

„ ,, r? ÖX dY dZ J . ,

X ,Y ,Z r^- , -r— , -r— respectively

u* v' W ÖU ' ÖV * ÖW

^ ^

X Fractional distance along blade, x = r/R

X etc. Non-dimensional f o m of derivative, x = X /p(nR)(7tR^) etc,

X etc. Non-dimensional form of derivative, x = X /p(QR)(wR^)R etc,

a

Angle of attack of blade element,

a = 6 - 0

fi

Instantaneous blade flapping angle, measured from

no-feathering plane

Y Lock's inertia number, y =

j

—

^1

6 Blade section drag coefficient

6 Instantaneous blade pitch angle measured from the

tip-path plane

X Inflow factor

-

-5-L i s t of Siynbols (Contd)

'^o' ^1 ''^°'

M ' ÏÏR

''•^°-\ ''•^°-\

=

HR

Al T> X- V C O S i }l Advance R a t i o , |ji = — j y ~ —

V Induced velocity through the rotor disc at any point ( r , ^ )

, , = J_ f rdr r

V M e a n induced velocity,

Vn, Value of the induced velocity at r = R in hovering, and r = R, f = ±2 ^ for\fard flight

V U n i f o m induced velocity

V Radial induced velocity distribution, — = - x* + 2 x V. Parameter in expression v.x cos \ir for the induced

velocity distribution due to forward speed p A i r density

7ïR b c

O" Solidity factor, cr = -rr

T

^ Blade azaiauth angle measured from the downvnjid

p o s i t i o n in d i r e c t i o n of r o t a t i o n

n Angular velocity of rotor

-6-1, Introduction

Mthough it is v/ell known that the induced velocity distribution through a rotor is far from uniform, little has been written concenaing the effect of this non-imifomity on blade flapping coefficients and rotor derivatives,

Glauert (l) suggested a triangular distribution of induced velocity from the front to the rear of the rotor disc, This distribution gave values of the lateral flapping coeff-icient b. agreeing more closel;/- with experimental measure-ments than values predicted using a uniform distribution,

Martin ("/) used the induced velocity distribution calculated by Iiangler (10), who treated the rotor disc as a cir'cular wing, to obtcln values of b. which compared favour-ably with flight measurements by Lïyers (13). ^ considering the effect of this greater lateral tilt on the force coeff-icients he concluded that there vrould be a significant effect on the rotor derivatives,

The Majigler induced velocity distribution was cal-culated on the assumption that the perturbation velocities due to the rotor disc v/ero small compared Vi-ith the free-stream velocity, It is, therefore,not applicable at low forv/ard speeds, (belov/ |i = 0,1 say),

To investigate the effect of non-uniform induced velocity at low foirward speeds a parabolic radial distribution has been chosen v/hich agrees vrell vd-th flight measurements by Brotherhood (9) on a hovering helicopter. To this has been added a distribution varying linearly from the front to the rear of the rotor disc and depending in magnitude on the advance ratio \i, Volues have been calculated for the flapping coefficients, the force coefficients and the rotor derivatives. These have been compared \rf.th values obtained assuming a \aniform distribution of induced velocity over the rotor disc, and with the results obtained by ivlartin (7) and the flight measurements by i^ers (13).

2, Notation

The British system of notation has been adopted i,e, all forces and moments arc referred to axes attached to the tip-path plane. The angle of incidence of the rotor disc is taken as being positive when the disc is tilted for-v/ard v/ith respect to the direction of flight. The system of axes is shewn in Pig, 1,

-7-The expression for the cyclic feathering of the blades vn.th respect to the ,tip-path plc^jae is

0 = A - A cos 1^ - B^ sin f ,,,., (2-1) v/here ^ is the azimuth angle in the plane of the disc and is measured from the dovmstream direction in the direction of rotation of the blades,

The expression for the blade flapping angle vd-th respect to the no-feathez'ing plane is

B = a - a. cos * - b. sin* + terms in '^ o 1 ' 1 '

higlier hanuonics ,,,»., .,.,,,(2-2)

It has been shovm by Lock (2) and others that, for the flapping and feathering systems to be equivalent, the first harmonic flapping coefficients are related to the cyclic feathering coefficients by the follovTing expressions,

^1 = ^1 1

-^ (2-3)

'

l =-^^1 ]

3 . The Flovv' Relative t o the Rotor Disc

For the r o t o r v/ith forward v e l o c i t y V, the component of V i n a plane p a r a l l e l t o the t i p - p a t h plane i s given by

l a Q R = V cos i ( 3 - l ) v/here \i = Y cos i / l R (3-2) i s knovm a^ the ' advance r a t i o ' ,

The v e l o c i t y perpendicular t o the t i p - p a t h plane i s

TvflR = V s i n i + v (3-3) v/here v i s the induced f lov/ through the r o t o r d i s c , and X

i s the 'inflow f a c t o r ' ,

-8-4. The Flov/ Relative to a Blade Elcnent

For purposes of estimating derivatives the rotor is asstimed to have a pitching velocity q. and. a rolling

velocity i>. Using the expression for cyclic feathering given by (2-1) the following expressions aj^e obtained for the velocity ccnponents relative to a blade element at radius r = xR,

(i) The velocity component perpendicular to the blade in a plane parjxllel to the tip-path plane,

U^ = (x + ti sin 1^ ) n R (4-1)

(ii) The velocity component perpendicular to the blade and to the cone surface

U^o = (a Li cos If" + X - -r X sin ^ - ^ x cos if')^ R

P o"^ n n

.(V2)

v/here a i s the base angle of the r o t o r cone,

( i i i ) The spanv/ise v e l o c i t y along the blade i s

(pi cos f - Xa )n R

The effect of this spanwise velocity is not consid-ered in the subsequent analysis since the dominant t c m [X cos f v/ill be small at law forward speeds,

The angle of incidence of the rotor blade element is

a = Q - 0

(4-3)

-1 "P ~ "P

where ^ = tan rr- ~ ~ since Up « U

Hence P _£_

'i. cos ifr- B.

a |i cos^fr + X - n X sin If' - Q x cos a = A - A. cos ifr- B. sin f

X + li sin If'

(4.-4)

-9-5* The Induced Velocity

5,1, The Induced Velocity in Hovcrinp:

Lïeasurements by Brotherhood (9) show that the induced velocity in hovering is far frora uniforr.i over the rotor disc. ffi.s experimental values agree v/ell vdth values calculated from propeller strip theory,

It was found that the induced velocity distribution, as measured by Brotherhood, could be approximated very closely (see Fig. 4) by the follov/ing simple expression,

V

•— = - x^+ 2x ,.... v5-l)

v/here v_, is the value of the induced velocity at the edge of the rotor disc and x = r/R. This expression represents a parabolic distribution varying from zero at the centre of the disc to a maximum value at the edge of the disc,

The folloT/ing integrals are nov7 evaluated for later reference. Note that X_, = v/Q. R , X = v /n R

ƒ X^ dx = I X^

I.

!

I,

X^x ÖX = 2- \^

K""^^

=

VÖ

\

X^x'dx = 1^ X^

^-.(5-2)

5.2, The Induced Velocity at Moderate ForiTard Speeds (li > 0.14)

Following Glauert (I) it v/as decided to superimpose an induced velocity distribution, varying linearly frora the front to the retvr of the rotor disc, on the induced velocity distribution in hovering, to account for the effect of fonvard speed. This linear induced velocity distribution is given

l o

-i n non-d-imens-ional for-i-i by

x^x cos V = \7^yx °°s If- .,,.,(5-3)

This represents an induced velocity varying linearly from a value -X^ at the front of the disc to +X. at the rear of the disc,

The choice of the value for X. is arbitrary. Glauert (I) suggested letting it ha.ve the same value as X which, in his paper, represented an induced velocity uniform over the vifhole of the disc. It v/as decided to let X. f X_,, for [X > 0.14 and later calculations of the flapping

coefficients a , a., and b, shov/ed good agreement v/ith experiiïiental values given in Ref, 13, and also with values calculated by Martin (7) using the slangier induced velocity distribution (see Pigs, 7-9).

The effect of the angle of incidence of the tip-path plane on the induced velocity distribution has been ignored since the incidence is small in practice ('Gyrodyne condition').

5*3* The Induced Velocity at Lov/ Forv/ard Speeds

At zero forward speed X. is zero and at moderate and high forv/ard speeds the choice of X. = X,- appears to give good agreement with flight measurements for the flapping coefficients. To cover the low forward speed range it v/as decided to assume an exponential increase, from X. = 0 to

\ = \f

given by

X^ = \j, (1 - e-^t^) (5-4) and to choose c such that X. = 0,9 Xn, for |i = 0,10,

This gives c = 23 and

X^ = X^ (1 - e-23^^) (5-5) Again this expression for X is somewhat arbitrary

but gives the proper end conditions (i,e. X. = 0 for n = Oj X^ £ \|, for li > 0.14).

-11

5.4. The Variation of X^p v/ith [>.

For hovering, the value of X_, may be determined from momentum theory,

The thrust T is given by

• R

• • ƒ

o

p27cr dr. 2v 2

P u t t i n g X = r/R and s u b s t i t u t i n g for v from (5-1)

1

T = 47tR^pv* / (x^* - 4!c' + 4x:') dx

pv* I ( x ' - 4!c* + 4x:')

•'o

v/hence ^ = ~ ^ %R^ pv^

T Nov/ the thrust coefficient C„ =

^

p^'i^R)''

therefore l"TT _

'^T = ^l44 S = 0.ö26^S (^"^^

The corresponding expression for unifom induced velocity is

•k^

= 0,707 ^JBJ (5-6A)

where ?^, is the non-dimensional form of the uniform induced velocity,

For moderate and high forward speeds Glauert (l) has developed the follov/ing formula for the thrust, by treating the rotor disc as a circular vmig of span 2R, and having elliptical loading.

T = (T^RV)

\

(5-7)

where V' is the resultant velocity at the rotor disc given by

_ 2 2 "^ -^

V' = | ( V s i n i + V n , ) + (V cos i) _| (5-8) and V is the mean induced velocity given by

-12-i-K

^m = - T / i- <3r f V dif (5-9)

T^ I J •'o * o

S u b s t i t u t i n g v = v (-x + 2x) +n RX x cos if- i n t o

1 (5-10)

(5-7) gives

% = 5/6 v^

°^ >m = n | = 5/6 \ p

(5-8) can be w r i t t e n as

V' = n R(X^ + li^)^ (5-11)

and by substituting (5-10). (5-11) and the expression for the

thrust coefficient in (5-7) the expression for X„ becomes

^C = f >>n = I ,: ^ , 1 (5-12)

L(^^i+>m) +l^*/

This leads to a quartic equation for X^, which

cannot be solved in general terms, Hov/ever for high

foj:n/ard speeds and low angles of incidence, i.e,

2 / N*

|i > >(|ii + >i|..|) , X_i is given by the simplified expression

\r = o»6 i;^ (5-13)

The corresponding expression for uniform induced

velocity is

Due to the difficulty in solving (5-12) for X^, and

also to the doubtful validity of this expression at low

forv/ard sxDeeds it v/as decided to use an empirical expression

for X„ of the form

\

=-^- (5-14)

B + li

and t o choose A and B t o s a t i s f y the follov/ing c o n d i t i o n s !

Xy = 0.826 VCy for li = 0 ^^_^^j

X^ = 0,6 C^ii for |i = 0,25

1 3

-A

and B are A

B

then

1

1

given by 0.6 C^ - 2.9 ^Tc^ 0.727 Vc^ - 2.9 VCj and \^ = 0.707 VC^ for n = 0 Xy = 0.5 C^|i for \i = 0.25 S^^^S 0.5 C. A' = ~ 1 - 2 . 8 3 VCj .(5-16)

S i m i l a r l y for imiform induced v e l o c i t y

K = - ^ ^ (5-14/V) ^ B' + n >(5-15A) ,(5-16A) 0.707 VC B» = 1 - 2.83V Cj

Curves of X^ and X,, against [x for a thrust coefficient C_ = .0055 a^e presented in Fig, 5.

5»5» The Derivatives of X_, X. and 7^^.

5»5.1. The Derivatives of X^p

\ - \ (i^f 0^) v/here C^ = C^(ii) therefore

dx^ dx^ axj dCj

d U = ai: " ac^ • a m (5-i7)

1 4

-aXm a 7 \ ^

vrtiere T — and r^^- are obtained by d i f f e r e n t i a t i n g ( 5 - 1 4 ) ,

3 li dCn givung A and öXj, 0.364 X,p 0.6(1-1.45 Vc ) - ^ Vc„ ^ S (1-2,9 Vc^)^BH.ii) * 5 «5 •2. The Derivatives of Xj,

The corresponding expressions for uniform induced velocity are

dXy aXy aXy aCj

dir = air •*• dc; • all

^"\j A' T where and d ji

'hi

(B'+ii)" 1

ac,

_{'T (1-2,83 Vc^) (B'+n)} 0.50(11,2*2 VC^) -(5-17A) (5-18/0 0.355 Xy .(5-19A) 5«5«3» The Derivatives of X 1

X^ = (1 - e"2^t')X^ = X^(X^,ii) where X^ = \j,(ii) therefore giving dx^ ax^ dXj ax^ d i i ~ a x ^ d|i "^ d \x d|i ^ ''d|i T Also ^ N ,, -23liN ^^T ac^ = ^^ - ^ ^ a c ; .(5-20) .(5-21) /6. ••.

-15-6, The Thrust Coefficient

The thrust T is given by the double integral

2% dx dT dx d If- . ( 6 - 1 ) cR dx i s 0 o The r e s u l t a n t f o r c e on a b l a d e element of a r e a ^R dP = - T - p a c n R f f T r l c d x ( 6 - 2 ) v/here U i s t h e r e s u l t a n t of ü„ and UT, and U -^U^ s i n c e

Urp > > Up,

Also the resultant force F is very nearly perpen-dicular to the tip-path plane so that

d T ~ d F = ^ p a c

"V C^

c dx

.(6-3)

Ely substituting (6-3), (4-4) and (V-2) in (6-1) the expression for the thrust coefficient becomes (sec Appendix l)

^T - 2

3 V 2 ^^ / 2 12

'T

2 * 4n

and for vmiform induced velocity

0 - ȣ

4n

.(6-4)

(6-U)

7. The Feathering Coefficients

F o r e q u i l i b r i u m of t h e r o t o r d i s c t h e c y c l i c

f e a t h e r i n g raust be such t h a t t h e aerodynamic moment p r o d u c e d on a b l a d e b a l a n c e s t h e dynamic raanent a b o u t t h e f l a p p i n g h i n g e g i v e n by

l.L = I n a - 2qfJl s i n ^ + 2pfil cos f , , , , ( 7 - 1 ) vdiere I^ i s t h e b l a d e moment of i n e r t i i about t h e f l a p p i n g h i n g e ,

The aerodynamic moment about t h e f l a p p i n g h i n g e i s / g i v e n b y , , .

-16-given by

dx

(7-2)

dP

Substituting for -r- from (6-3) the following expression is obtained for M. (see Appendix II)

M [^ = ^ pacn R / z . A -D liB. u'A \

^ l o ' ^ 3 4 ^ 6 n

3

4

/

/ p L i ^ 5 ^ - ^ 1 2 , 3 ZT, ^

^ *

V~fi"

2 - 1 2

^^^T

- r ^ 3 ^^o - 8 ^^ ^ i ;

+ sm / na^ X^ p A^ ^ , + cos ^ \^- - y - - — + — ^ - ^ - Q M ^ i

+ terms in higher harmonics I . Comparing (7-1) with (7-5)

.(7-3)

a |iB, 2 4 ^ ^ ^ 10 '^T 3 3 bn I ''' ^^ A ^ = -

4

1+t^*

3 + 4 " 4 . "^ rn

.(7-5)

B. =

4

1-4 1^*

13 ^^^o " 2 " 12 ^ \ "^ 40 ^ yn J

_(7-6)

V/here y = ^—r— i s known a s L o c k ' s i n e r t i a number,

The c o r r e s p o n d i n g e x p r e s s i o n s f o r liniform induced v e l o c i t y a r e

^ 0 . / ) . ^ . ^ . ^ . ^ l ..(7-u)

-c=i

A ^ =

4

1+11^" _ 3 " 40 •*• yfi

.(7-5A)

^1 =

4

ulx'

3 ^ o

-'^-^hj^h*^] (7-^'-)

/ 8 . . . .

-17-8, The H Force Coefficient

The H force is the drag force in the tip-path plane. From Fig, 2

dH = (dD cos 0 + dL sin 0)sin f - (dL cos 0 - dD sin 0)sin a cos ifr (8-1)

Noi"/ a and 0 are both small angles so t h a t

d H = d D s i n ^ + 0 d L s i n l ^ - a d l c o s ! ^ . . ( 8 - 2 )

The term a 0 dD cos ^ is neglected since a and 0 are both small and dD is small compared v/ith dL,

Now dL = i pc C^ U^ dr (8-3)

v/here C, i s the local blade l i f t coefficient = a(e-0)

h

and dD = -^ pcöU* dr (8-4)

where 5 = blade section profile drag coefficient, assumed constant.

Substituting (8-3) and (8-4) in (8-2) and putting U = U^ and 0 = UpAj^

'

B(~

Up/

M I

f

M 1

dH = -I pc U,j, { 5 + a :j5- (0 - :^J sinif - a^a \0 - ~ y cos if J R dx (8-5)

Martin (7) neglected terras involving 0 but

retained such terms as 0Ö, a 2( and a 0. Since 0, 0 and a are all of the same order this simplification v/as not

considered to be justifiable, and the terms involving 96* have been retained,

The H force is given by the double integral

^-h 1 ^r§^*

(8-6)

o o

Substituting (8-5) in (8-6) the following expression /for the ,.,

-18-H

_{is obtained}

p;ca(aR)'

'\ ^r. \^^r. ^J^^ for the H force coefficient 0,.^ = ~

(see Appendix I I I )

ao- I 6u ^o f . 2 ^ V "^1 / . 5. ^ \ ^^I'^o """o ^b'^1

°H = T j j t •*• T " (^'^ + 3 Xj) - — (til + -^ X^) + -J- + i ; - + — ^

jiA.X. p / c ^ ^ ^ I

— r r + 2JT v^"^ •" ÏÏ '^T • F "• 8 ^'^lyj •••(8-7)

The corresponding expression for uniform induced

v e l o c i t y i s

^ H - 2a

~~ liii B. A a lia

•^20 V^ + \ j - 3^ - c ^^^1

(8-7A)

9, The Side Force Coefficient C

_YS

Prom Fig, 2

dY„ = - (a dL sin^ + 0dL cos if + dD cos if ) 'S ^"o

Substituting as in expression for H force

(9-1)

dYg = - i

_{P°UT [ I %^ (^ - Ü;) f}

s i n If +

_J

Rdx

(9-2)

Performing the double i n t e g r a t i o n as before gives

(see Appendix IV)

acr o

YS 2 2

3|i^i + 2iiX^ + B

r

1 (3 ^ ^

''-h^

X,

+ 2^

(lii

+ I X^)

+ ^ (^1 + ^ x^) - -g TT

-

aT \J

^^B,

-2n V^^^ + f ^ : - r - ^ - r / J - " ^ 9 - 3 )

The corresponding expression for uniform induced

-19-velocity is

Cyg - -

f- [ / |3ix'i + 3^\j + B^ ^ + li'j- I

i^^i^

(t^i+^j)

P_ 20

10» The Rotor Stability Derivatives

The rotor derivatives of importance, for the case of zero flapping hinge offset,

are,-(i) The force-angular velocity derivatives

\ ' \ >

yp

(ii) The force-velocity derivatives X , z , y , X and z

U ' U ' *^V' W V7

Russell (6) and others have shown the b a s i c

equations for estimating r o t o r d e r i v a t i v e s to be

AX = - TA a^ - AH , . . . . . . ( 1 0 - 1 )

AY = TAb^ + AYg (10-2)

AZ = HAa^ - AT (IO-3)

These relations follov/ immediately from Fig. 3.

For the case of controls fixed a cliange in longi-tudinal flapping A a. results in a change of incidence of the disc A i = - A a. i.e. -r— = - 1 = ^rr .

1 oa. oD.

1 1 In estimating the rotor derivatives the change in induced velocity in the disturbed motion was tal-cen into account. This v/as done by assuming equations (5-17) and (5-18) to apply in the disturbed state. This assumption seems reasonable provided the disturbed motion takes place slowly,

-20-In the expressions for the derivatives, in the

following sections, A. is replaced, by -b. and B. by a., from (2-3), Equations T/ith the suffix 'A' refer to the

uniform induced velocity case.

11, The Force-Angular Velocity Derivatives x , yp and z

The force-angular velocity derivatives follow from equations (IO-I) and (10-3)

(-°T

aa

J.

dq a q/ ,(11-1)

n

_{( 0 .}

ab^

aCyg

ap

• ) ,(11-2)

z =

q.

. (c

da H aq" 1

ac^

a^.

:) .(11-3)

Cm a-nd C„ are obtained from equations (6-4) and (8-7) respectively. The expressions for the partial deriva-tives are

ac^

F P

ac^

aT

aCy li acr

8n

= 0 ,(11-4) ,(11-4/^) , ( i i - 5 ) , ( i i - 5 A )

Ü0 jfli. i:t ! ^ f r

a p - i2fi " 2 0 aCj • a p ,(11-6)

a p ~ i2n " 6 ac^ * a p

, ( I I - 6 A )

2 1

-da

o

a q

da^

aT

aa^

aT

aa,

1 - y

^ ^ ax^ ac^

n ' 3 ^ • a p

, ( i i - 7 ) , ( i i - 7 A )

, (11-8)

1

1 - è^

16

- - 2 ,

axy

ac„

acj • a P

^ '^ " yn ( 1 - ^ ^ )

ab^

aT

ab^

a~i

1

4n^

JT

ü Üo 1 fN ^ ^ J t

3 ö p "^ 4 ac,

T

a p y

Y Ï Ï "

4

1 4ii^

aa ,

• C M • I I I , 1 . ^ a w t a a

3 a p yn

, (11-8A)

,(11-9),(11-9A)

, .(11-10)

, (11-10A)

....(11-11),(11-11A)

( 5 - 1 9 ) .

ac,

o (1 + ^ ' )

aXrp/aCn, e t c . are obtained from equations (5-18) and

""H

a q

acjj

a q

^^YS ap acr ~ 2 acr - 2 acr ~ 2

aa^ ^a^ ^^L^ ^ _ 1 . ^ ] ! ^ ( ïN ''o

_ y _ __ - ^ til - 24 V " a q \ ^g - ^

a a, /lia n*A

_{a ab.}

2 ~ t ^ ^ " t ^ j y " ' S " a T

_ J . ( ! Z i "* "o 1 . 1 , 1 "o ""1

a q \ 4

•aa

..(11-12)

..(11-12/0

f l i ^ i + liX™ + T - ( T + ^ * ) - f ti^i T " 2 ^3

aa, 1 a

_{1 o / I}

4 ^^oj " a p |_2 ^3,

Ö ^ ( i - 2 H )

_ L .

_{16 ^^^1 •*• " T j}

^•b,

- aTV 4 ^^^ ^ 24 V •" TT H -^ 12 N - 24 ^i^ -^

a p \ 2 "^ 12 ^T ~ 6 "^ 4 / " 2n \ 3 " 8 / ^T^-T^^

2 2

-aa

YS _ o^rf

aa

aa. a

V4 ' ^ 4 / ' ^ drj\2^%- h-J - 2n \3 QJJ

(11-13A) | - 2 t i « )

ab,

_{_1}

dp

12» The force-Velocity Derivatives x and z

•^ u u From e q u a t i o n s ( 1 0 - 1 ) and ( 1 0 - 3 )

aa. ac„

u ~ '•'T a li d [x aa^ a c ^ u ~ H a n ~ a i i . . • • • < , ( 1 2 - 1 ) . ( 1 2 - 2 ) The e x p r e s s i o n s f o r t h e p a r t i a l d e r i v a t i v e s a r e aCj . ^ / . . dX„ a |i " a cr 1 + ^ acr

r v i ( ^ ^ ^ o - | - f 2 i f ) ^''-'^

24

ac, _T aCy

aT

aa

o X.

a |i "" 2 a cr 1 +

_4^^acr

(^^^o-|4 a i )

(^^-2 " 1 0 d i i " 3 * 3 3A) , ( 1 2 - 4 )

aa

o X, a ji ~ 2 liA 3 d li

i

3

' . . . « O '^"*4A^ ""1 a }i aa^ a li

ab^

=

=

1

1

4

4

4.^

4

dX„ a li 1 - ^ ^

f ^^o - ^^^ - fe ^! - fe i^ dT " I ^^

2 1 1 ^ \ j 3

^ A^ - Iii - 2

NJ

- 2 t^ d U - 4 '^'^l

a da . dX. o li . o 1 1 -u 3 3 a i i 4 d n "^1 . . . . ( 1 2 - 5 ) . . . . ( 1 2 - 5 A ) . ( 1 2 - 6 )

2 3

-ab^

a |i

5i

a V

uh'

a öa o li o , 3 :> a li ^ 1 ,(12-6A) B.0- \<

" 2 C

k -^ ^0 V'^ ^ 3 ^T

a . i 1

T " - ^ 4

2 "o 9a o 1^^^ '^-1 o 1 1 1

~ + r - r / -^ aT

aa. /na

li A

4

— 2 . H i > - ^ X 2 4 24 T

ab,

li>, 1 o •^ a li \ T ^ " "T"/"^ dji dX^ / liA^ 5a^ dX^ / a^ ^.b^

24/"^ d^i \ ^ "^ TS"/ J

(12-7)

^°H ao-fö „ / . ^J^ %

— = r 2^ * ^^oU^-*- — ] + - + — •*• a (i \ 2

a . i da / u a b . 1 o / ^ o 1 9 /i \ 4 % ab^ dXy / IXA^ a^

9a, / ' ^ ^ ^ ^ i i i \ j 2 " 4 " 4

a|i dii

4 / J

. . . . • . • • f « . . v 1 ^ — I A ^

13» The Force-Velocity Derivatives x and z

*' W W Praïi e q u a t i o n s ( 1 0 - 1 ) and ( 1 0 - 3 ) da X = O R I - C^ T-^ w T aw

!5i

a w ,(13-1)

r ^^1 ^°T

w , H a Y/ a w . ( 1 3 - 2 )

The effect of a disturbance velocity v/ in the positive z direction is to cause a uniform flov/ v/ through the rotor" disc in the negative z direction. The inflov/ through the disc then becomes

X nR = nR (|ii + X + X.x cos If ) - w

o r n o n - d i m e n s i o n a l l y

X = Lii + X + X, X cos If - X

'^ o 1 ' v/ . ( 1 3 - 3 )

Ï£CHNISCHE HOGESCHOOt.

VLIEGTUIGBOUWKUNDE Kanaalstraat 10 - DEU'J

2 4

-v/here

_{\ / - QR}

.(13-4)

It follows, therefore, that

dw

J

nR ax

v/

1 a

QR a(|ii)

,(13-5)

Equations (13-1) and (13-2) may then be v/ritten as

da

w - C,

1

dC.

H

T ax w ax_.

v/

.(13-6)

where

aa

₁

¥/

ax

^H ax

v/

ac^ ax" w

v/

a

aTI)

(13-7)

.(13-8)

The relevamt p a r t i a l d e r i v a t i v e s are

ac^ ax"

v;

ac,

acr

4

T •— dX

ag-4

T/

li ^ ^ ^

j 24 acy dXy 1^ acr

,(13-9)

,(13-9A)

aa L( ?• ax„ ac-,

o X LI X. Ï Ï

ax " 2 3 ~ 10 ac_, • ax

w L T v^

,(13-10)

da

c

ax w

1

2

1 1 fin f^T

3 • 3 acy • ax^ , ( 1 3 - 1 0 A )

da.

1

dX w

da^

dx" 1 -h 4

w

1 -ivi'

l . , 2 2 " 12 ^^ dc^ • ax T w

ji 1 f^J f S

2 ~ 2 '•^ dC ' a x T v/ ,(13-11)

,(13-11 A)

2 5 -db, 1 dX 4 w

ab,

1

ax

vr

1

-^h'

1 H4li'

da , ax, ac^,

ji o _1_ 1_ _ T

ax._ "^4 ac„, • ax ,

3

v/

aa 1

T y 1 dX dC H dX v/ w acr 2 a, P^A 'W ! da c

ax

w

r^

l^a. ₁ ( ^ -11* A

£ - ü i _ ^

24 T ab^ / | i X ^ _a

ax \ T5~ ' T"

w ^

ax^

ax"

w .(13-12) , ( 1 3 - 1 2 A )

(^-1x0 *^(^^^)] (^>^3)

ac.

_{H acri ""l}

ax

_?/ ^^

4

da c

ax

v/

aa^

9 \ |ia

^ \ 4 " 2 " 4 " 4 ^ J /

r

a db. o 1_

F" ax

w 5Xy / li/.

ax

_w

4 / -J

.(I3-I3A)

14» The Force-Velocity Derivative y

A velocity v in the positive y direction causes the H force vector to rotate through an angle v/V cos i giving a component -Hv/V cos i in the y direction. In additican there is a change in the lateral tilt of the rotor

disc^i b. = - a. — r giving rise to a force -Ta. r; r

1 I V c o s 1 e» e I V c o s 1 i n t h e y d i r e c t i o n , T h e r e f o r e - ^ = - T^ r ( H + T a . ) (1A^1) V V c o s 1 ^ 1 ' ' whence y ^ = " ^ ("^H •*" °T ^ ^ ^''^"^^ This e x p r e s s i o n i s n o t a p p l i c a b l e f o r t h e h o v e r i n g c o n d i t i o n where li = 0 , b u t b y symmetry i n heaver i n g

^ v = ^u (^^-^)

-26-15. Calc3ulation of Force Coeff icicnt^', Flapping Coefficients and Rotor Derivatives for a !Ihrpic;rl Case

Values of force coefficients, flapping coefficients and rotor derivatives have been calcnilated for a lypical case using values given in ref, 13. The details of the configura-ticjn are given in Appendix V.

Values have been worked out for both tmiform and non-xmiform induced velcDcity distribution. The results of the flapping coefficients at moderate forward speeds are comr-pared vriLth results calculated by Martin (7) using the i.isngler

induced velocity distribution, and v/ith flight measurements given in ref. I3.

The results of the calculations are presented as

f o l l o v / s , -F i g . 7 . ' 8 . ' 9 . ' 1 0 . ' 1 1 . ' 1 2 . ' 1 3 . • 1 4 . ' 1 5 . ' 1 6 . ' 1 7 . ' 1 8 . • 1 9 . a 0 ^1 ^ %' ^W X q y p z q z u ^u z w \ y.r v s v s v s a^,l3 °YS v s v s v s v s v s v s v s v s ^ 1^ (^ 1 1^

t^

^i ti li

n

V-(p. ( ' (• v s v s (ü (•

r

(• (• ( ' ( ' ( ' = y-IJ = 0 . 1 4 - 0 . 2 4 ) ' ) • ) (ti = 0 - 0 . 1 4 ) (• • • ) 0 - 0 . 1 4 )

/16, . , ,

-27-16, Discussion

Referring to Pigs, 7 - 9 it can bo seen that the flapping coefficients, as calculated from the induced velocity distribution adopted, give good agreement v/ith the flight measurements of ref, 13 and Martin's results (7), based on

the Mangier induced velocity distribution. In particular the values of the lateral tilt of the disc, b^, compare favourably, whereas those for the \:iniform induced velocity distribution considerably underestimate the actual case.

The values of a., the longitudinal flapping coeff-icient are vinderestimated by all three theoretical induced velocity distributions. This is due to the fact that no

account is taken of lateral asymmetry of the flow through the rotor disc. Certainly such asymmetry must exist since the effect of cyclic blade feathering (and/or flapping) is to produce a different lift distribution over the retreating blade than over the advancing blade, ' Hov/ever at low forv/ard

speeds this difference vri.ll be small and its effect on the induced velocity distribution can probably be ignored. At higher foryrrjcd speeds it could possibly be taken into a>.ccount by introducing a term X9 x sin if into the expression for

the induced velocity, v/here Xr, would be a function of the advance ratio li. It v/ould probably be difficult to find an expression for Xp(ii) analytically, but an empirical expressicxi based on experimental results might v/ell be used.

It is doubtful if the expression adopted for the induced velocity actually represents in any detail the true flov/ distribution through the rotor disc, except at or very near the hovering state, Yilhat it does represent is the overall trend of an increase in induced velocity from the front to the rear of the disc, which has been observed. This appears to be sufficient for the estimation of flapping

coefficients and hence also of rotor derivatives. The Mangier induced velocity distribution, on the other hajid, probably gives a much truer picture of the details of the flow through the rotor, i&asiorements by Pail and Eyre (II) and by

Palabella and Meyer (12) appear to confirm that the prediction of upflov/ over a region of the forwea-d pert of the disc is correct, Hov/ever the Mangier distribution involves somev/hat complicated expressions and it v/ould appear that the much simpler representation of the flow used here is sufficient for the purpose of estimating rotor derivatives,

Fig, 10 shows the values of the flapping coefficients over the lov/ fcarv/ard speed range, a. is the same for both uniform and non-imifom induced velocity, b. is much greater

-28-for the non-tmifonn induced velocity distribution because of the term X./4 v/hich talces account of' the longitudinal asymmetry of flow through the rotor disc. A. is slightly smaller for the non-uniform induced velocity case indicating that the resultant aerodynamic force acts closer to the blade root than for the iiniform induced velocity case.

Pig. 11 shoi(7s the variaticjn of the drag force coefficient Cji and the side force coefficient Cy.„ with |i fc3r the two cases. It is interesting to note that C„ is somev/hat smaller for the case of non-uniform induced

velocity than for the case of uniform induced velocity. This is due to the term A. a /6 being greater in magnitude than the additional terms involving X.. C„„ is negative for both cases but is considerably greater in magnitude for non-uniform induced velocity. This is due to the larger values

of A. - = b. and also to the terms involving X.»

The force-angluar velocity derivatives are shown in Pigs. 12-14. The derivative x is the sane for both cases in as much as C_ ^ l/aq is the same and the

contribu-dC

tion from B/dq is snail and very nearly the same. y^ is also •unaffected by non-uniforn: induced velocity since Cj ^^1/dP and ^^YS/dP are virtually identical for the

two cases. The derivative z is slightly different for q o -^

uniform and non-uniform induced velocity. It is proportional to C„ since da./aq is the same for both cases and

ap

T/aq = 0 . This derivative is exceedingly small and v/ould probably be ignored in most stability calculations.

With regard to the force-velocity derivatives it can be seen frcan Pigs. 15 and 17 that z and z are

^ u w v i r t u a l l y the same f o r uniform and non-uniform induced v e l o c i t y . The expressions for dC^dn and aC_/aX are very n e a r l y the same for the tv/o cases and the C„ " l / a n

and C„ 1/ax c o n t r i b u t i o n s t o tliese ' z ' d e r i v a t i v e s are n e g l i g i b l e .

The d e r i v a t i v e s x and x are a l s o v i r t u a l l y

u w ^ i d e n t i c a l for uniform and non-tmiform induced •'/elocity. The

Cm l/a|i and C„ l/dX terms are dominant i n the

-29-expressions for these 'x' derivative? so that the small changes in dC-Vdii and aC„/aX. for the two cases are relatively unimportant.

The derivative y is also very nearly the saine for both uniform and non-uniform induced velocity. The dom-inant term in the expression for y is Gma. v/liich is identical for the two cases. The small differences in C„

n have little effect.

Summarising it can be said that the only derivative appreciably affected by non-uniform induced velocity is z which is very small and relatively unimportant.

It appears that, at low forward speeds, non-uniform induced velocity has no significant effect on rotor derivatives. At higher forv/ard speeds it is possible that its effect might be more significant. Certainly if a lateral asymmetry of flow through the rotor disc v/ere taken into account the values of a. and its derivatives v/ould be different for uniform and non-uniform irxdaced velocity. This v/ould affect all derivatives to some extent and particularly x , x , x and y , For a highly loaded rotor at high f orv/ai'd speeds it would be expected that C,T would be larger relative to C™

than for the case of the lightly loaded rotor at low foxTivard speeds considered here. This would mean that the

C,j. l/du. and C„ l/ax contribution to z and z

H ^ H ' w u v/ would be significant and the effect of non-vmifom induced velocity might be important. There is some doubt about this last statement, hov/ever, for at high for\/ard speeds and high disc loadings, the main contributions to C„ would probably come from the \xA and lia* terms v/ith the result that C„ would be very nearly the same for both uniform and non-iinifortii induced velocity,

-30-17. Conclusions

1) An important effect of non-unil'orm induced velocity is to increase considerably the magnitude of the lateral flapping coefficient b.,

2) The value of C„ is somev/hat less fc-r the case of non-uniform than for unifom induced velocity and the value

of Cy.„ considerably greater,

3) The effect of non-viniform induced velocity on rotor derivatives at lov/ forward speeds is almost negligible except in the case of z v/hich is a very small derivative,

1, Glauert, H, 2, Lock, C N . H , 3, Squire, H.B. 4, Zbrozek, J.K. 5» Simon, H.A.

6* Russell

7» ifortin

8, tlafner, R. 9. Brotherhood, P. REFERENCES

A general theory of the autogyro. A.R.C. R. and M. No. 1111.

Further developnent of autogyro theory. A.R.C. R. and M. 1127.

The flight of a helicopter. A.R.C. R. and M. 1730.

Investigation of lateral and direc-tional behavic3ur of a single rotor helicopter (Hoverfly).

A.R.C. R. and M. 25O9,

Unpublished College of Aeronautics Thesis,

Unpublished College of Aeronautics Thesis.

Unpublished College of Aeronautics Thesis,.

Rotor ^sterns and Control Problems in the helicopter.

Anglo-American Aero. Conf. 1947. An investigation in flight of the

induced velocity distribution under a helicopter rotor v/hen hovering, A,R.G, R. and M. 2521,

-31-10, Mangier, K.ïi"., The induced velocity field of a rotor. and A.R.C. R. aiid LI. 2642.

Squire, H.B.

11, Pail, R., and Downwash measurements behind a 12ft. Ejre, K, diameter helicopter rotor in the 24ft,

wind tunnel, A.R.C. 12,895.

12, Palabella, G. and Determination of inflov/ distributions Meyer, J.R. frcm experimental aerodynamic loading

and blade-motion data on a model heli-copter rotor in hovering and forv/ard flight.

N.A.C.A. T.N, 3492.

13, ¥ïyerSf G.C. Plight measurements of helicopter blade motion with a comparison betv/een

theoretical and experimental results, N.A.C.A. T.N. 1266.

3 2 -AEFENDIX I D e r i v a t i o n of T h r u s t C o e f f i c i e n t f o r Non-\inifonj Induced V e l o c i t y 2% J 2% dT dx -T— . dif dx o o P r a a ( 4 - 1 ) and ( 6 - 3 ) T a r / dx / / • ..^' .3 .t / ( x + li sinif; a d \f iT^

s =

_{pr(R{<^Rf ^"^ Jo -o}

S u b s t i t u t i n g f o r a from ( ^ 4 ) I n t e g r a n d ,-.

=(x+ii s i n If) A A. c o s V^B. s i n If -' o 1 1

a iicos\f+iii+X +X.xcosif-rrxsin\p~Q x cos\f X + n sinif

P

= - (x+n siraif) ( a \x c o s f +lii+X +X.x cosif - jpc sin^f - Q •'^ ° ° ^ ^^ + (x^+2 |ix s i n If + i i ^ s i n ^ ^ ) ( A - A cosif - B.sinif )

. p 2 2 2 . 2 0 2 2 \

= Sin !f ( — X -li i-iiX - B . x +2iiA xj + cos ifr(-|ia x-X.x + -^ x - A.x ; + s i n If cos if (-H a -liX.x+n § x - 2iiA x) + s i n (f (|i — x -2|iB x+|i A ) - li B . s i n \f -li A . s i n f c o s if - X x - | i i x + A x _{1 1 o o} /•I * 3 V ' ^ 2 * ^ / ' * 2 " 1 2 ^ " 2 ^ l^ dx

°T - 2

/Appendix I I . . .

3 3 -AI^EEI'©IX I I D e r i v a t i o n of E x p r e s s i o n f o r M, A 1 M, = / X 1 ^ d x A J dx o Prom ( 4 - 1 ) and ( 6 - 3 ) ' 1 4 / 2

= p- pacR / x(x+ii s i n if) a dx

^^A

" o

Substituting for a from (4-4) Integrand

2 2 p 3 2 9 2

= - X x^-Hix +A X + sinf- (fj X - ^ ix-liX x - B . x +2iiA x )

+ cosif (-|ia X - X . x + •^ X -A.X )+ sin^^ c o s f (-^ a x-iiX.x +li -^ x -2iiA.x ) + s i n If (|i r- X -2iiB.x +ii A x) - n B . x s i n \f-li A x s i n if cos if

Now s i n (f = i' - i" cos 2 if

sin^*- = -f s i n if - r- s i n 3 '^ 4 4 , 2 1 1 s i n if . c o s ^ = 7" cosif - -r c o s 3 ^ 4 4 so i n t e g r a n d - 2 . 2 3 uP 2 ^ 2 2''^o = - X^x - l i i x +A^x + 2n •'^ -I^B^x + li - ^ X

+ s i n !f( — X -li ix-iiX x-B.x +2iiA x - r |i B.x)

Q, o 1 ' ^ o 4 ' ^ 1 '

+ cos i^(-tia x^-X.x + -^ x-^ - A.x - T ^ A.x) + terms in higher harmonics

T h e r e f o r e „

„ 4 / T . A _ i i B . i i A \

/ p li i 5 1 2 5 2 + s i n i f \ ^ - ^ - ^ liX^ - ^ + - liA^ - -^ ü B^

-34-(

^^%

N p ^ 1 ^ ^

+ terms in higher harmonies.

APPENDIX III

Derivation of C„ for Non-Uniform Induced Velocity

H =

2% d.f

Prcm ( 4 - 1 ) , ( 4 - 4 ) and ( 8 - 5 ) p _ 5E

^H - 47C j dx / (x + li s i n f)

g / l i a COS if+|ii+X +X.X cos if - -j^ x s i n i f - "fj x cos f

[

!^(-

_{X + li s i n lf} )

(v

L ( A - A , o \ o 1 A cosif B sinif c o s lf B s i n lf

-|ia cosif +iii+X +X.xcosif - ^ x s i n if- ^ x cosif X + li s i n lf

lia c o s lf+iii+X +X.XCOS lf - J7 x s i n i f - j j xcos if j X + |i s i n ^

s i n lf

cos lf dif

I n t e g r a n d = _{- ( l i a cos ^+iii+X +X.xcosif - —xsin\f- ^ xcosif ) s i n ^} p „ 2j

+(x+iisini^) (na c o s ^ + i i i + X +X.xcosif- —xsin^ - ^ xcos ^ ) ( A - A c o ^ - B . s i n i f ) X s i n ^

+ a (lia cos lf +iii+X +X.xcosif - — xsinif - ^ xcosif ) cos if

"^

2 2 2

+ (x +2iixsin lf+ii s i n *) — s i n ^ - a ( A - A . C O S i f - B . sinif ) c o s if a o o 1 1

r-* 2

O 3 9 ? 5 2> 5* "D 2 2 <-« o o

- ( n * a c o s if+ii i +X +X.X cos if+ — x s i n if+ ~ x^cos if+2ii a i c o s if + 2|ia X cosif

-35-2 P Cj -35-2

i-2X.iia xcos if-2|ia — x s i n if cosif -2iia ;?xcos ^+2iiiX +2iiiX.xcosif

1*^0 ^ o n o Q " ^ o l

- 2iii — X sinif Q

q P a P 2

- 2 u i -* xcosif +2X X.xcosif -2X — xsinif -2X •^ xcosif -2X. — x sinif oosif n 0 1 o Q o Q 1

-2X. -TC cos if+ —^x sinif cosif sinif

1 n

n^ J

+ (x+iisin^ ) (lii+X ) ( A sinif -A sinif cosif - B . s i n i f )+(X.x+iia - -^ x)

2 2

(A s i n lf COS ^ - A . s i n ^ c o s «f+B. s i n ^ cos V' ) - ^ ( A s i n i f - A . s i n if cos - B . s i n if )+li a cos^if+|iia cos^f +a X cos^^

5, a P il q ,

o o 2 +a X.xcos»' - x s i n i f cosif - - r r r x c o s i f

o 1 Q n + ( X +2iixsinlf +ii s i n lf )(— s i n i f - a A cosif +a A.cos if+a B.sinif cosif )

^ • ^ '^ ' ^ a o o o l o l '

, 2 2 2 8 2 N / . ^ A 2 S

= Sin lf (-u 1 -X - 2iiiX +iiiA x+X A x+ — x J+cosif ( a uix+a X x - a A x ; _{^ ^ o ' ^ o ^ o o o a ' ^o"^ o o 0 0 '} + s i n ^ cosif (-^li^a i-2iia X -2iiiX.x+2iii ^ x-2X X.X+2X ^ x-iiiA.x-X A x

2 q 2 P 2 2 2 \

+ A X.x +A •* X +ua A x+a — x +ii i a +|iX a +a B.x -2iia A x ; o 1 o n ' ^ o o O n o ' ^ o o o l o o _{n ; o o o Q}

. B > Y

-n

2 y * P "P T * 3 2 5 \

+ s i n lf (2iii — X+2X rr x-B.uix-B.X x - — A x +A u i+A |iX +2 — |ix) ^ • ^ n o Q 1*^ 1 o n o o*^ 0*^0 a"^

2 2 Z n Z 2 \ ' Ê "P P P n 2

+COS if(iia x+a X.x - a ;^ x +a A.x ) + s i n ifcos if (2iia — x+2X, — x - —^ x ^ • ^ o o l o n o 1 ' * o ü i n Q2

P 2 2

- X . B . x +B. X X -lia B.x+A. — x -li A.i-uA.X +iiA X.X-LLA ^ x+n a A

1 1 i n ' ^ o l i n ' ^ l ' ^ l o ^ o l ^ o Q ^ o o

2

" ^ % n x+2iia^B^x-ii^a^A^)+sinl'' c o s ^ (-(i^ a^-X^^x^ - ^ x^-2X^iia^x+2|ia^ ^ x

q 2 2 q 2 2 2 Q .

+2X. ^ X -X.A.x +A. ^ X - u a A.x+u a +iia X.x-ua ^ x+2iia A.x) i n 1 1 i n " ^ o i o c i o n ' ^ o i

2 2 2 2 3 Q

+ s i n ifcos ^(-liX.A x+|iA^ f- x-ii a A.+ii a A. ) + s i n ifcoslf (-nB.X.x+iiB. ^ x

^%^ +(iA^ § x - i i \ ^ B ^ + ( i ' a ^ B ^ ) + s i n % ( - ^ x%B^ f x'-ü^B^i-liX^B^-üA^ § x + ü ' f )

n-^

P 4 +liB^ - X s i n \ Now /2^ / ( c o s ö , s i n 0 , s i n ö c o s 6 , s i n ^ ö c o s 0 , c o s 0 s i n 0 , s i n 0 , s i n 0cos0) d0 = O o

^-^

P 2 2 P 4

/ ( s i n * 0 , c o s * e ) d 0 = %, j s±n 0 . c o s 0 d0 = ^ , J s i n 0 d0 = - ^ • O r\ r\ T h e r e f o r e ^H

-^L

B

1 ..

B.

C^^ n ^ •" ^o 75 ^ - F" t^^ - F- V - tn V"-^ 2

, A li i |iA X 2. o^ . "^ o o r lia^x a X.x* a ^ q x ' a ^X.A.x

* 8 l ' ^ l l ^ j ' ^

F i n a l l y A.a^ lia" + ~6 Ï T - + 20 (^^^ -^ t ^T - r •" 8 ^'^1) /Appendix IV • , ,

3 7 -iiPDElKDIX TI D e r i v a t i o n of G^^. f o r Non-Unifom Induced V e l o c i t y 1 ^^

\ -

2.

J^^ 1^

ax •

Prcan ( 4 - 1 ) , ( 4 - 4 ) and ( 9 - 2 ) 1 2%

Cyg = - • ^ ƒ ^ I (x + t^ s i n if)j a^ f A^-A^cosif -B^sinif

lia cosif +iii+X +X.XCOS if- -^ xsinif - — xcosif

sinif

p X + li sinif

g /[xa. cosf +lii+X +X.xcosif - rr xsinif - q / n xcosif \ — X + li s i n ^

/ lia cosif +|ii+X +X.xcosif - -jscsinlf - •^ xcos A j -\ U - A ^ cosif- -B^ sinif 2 _ J ii ii ^yjcceif kv

X + li s i n ^

'iv - — xcos»" ^j ^

X + n sinif - -^ P Q N ^

I n t e g r a n d = - ( | i a cosif +|ii+X +X.x cosif - -r xsinif - Q x cos'f ) cosif + (x+usinTf ) j - a (ua cosif +|ii+X +X.xcosif - — xsinif - 7; x c o s ) sinif

^ l o ' ^ o o l n n

+ (lia cosif +iii+X +X.xcosif- 7^ x sinif - % xcosif ) ( A - A cosif - B . s i n ^ )cosif'

^ • ^ o o l n u ' ^ o 1 1

--2 --2 --2 1 ^ ï

+ (x +2iixsinif +ii s i n iM a ( A -A.cosif - B . s i n i f )sinif + — cosif [

f o o i 1 a j

,— 2 2

I 0 0 2 2 2 2 S 2 2 TD 2 2 2 2 2

= ' - ( u a cos if+u i +X +X.X cos if+ —- x s i n lf + ^ x cosif +2ii a icosif

l ^ ^ o ' * ^ 0 I ' ^ n n^ o +2iia X cosif

^ 0 0 *

2 p 0 2 P +2X..lia xcos lf-2iia — xsinif cosif -2iia - ^ c o s if+2iiiX +2iiiX.xcosif - 2 i i i -^ xsinif

a P a P 2

- 2 i i i ^ xcosif +2X X . x c o s ^ -2X ?r xsinif -2X fr xcosif -2X. 77 x sinif cosif '^ n o 1 o n o " 1 **

o 2 2 Po i

-38-+ (x-38-+iisinif )j - a lisinif cosif - a liisiiiV- - a X sinif - a X.xcosif sinif

P . 2 q . , ,

+a — X s i n lf + a -r x s m if c o s if

o n *^ o n

+ (lii+X ) ( A cosif -A.cos^ if-B.sinif cosif )+(lia +X.x- -^ X ) ( A cos i^-A.sinif c o s if

o o 1 1 o i n o 1

2 . - B s i n f cos ) - — X ( A sinif costf- -A.sinif c o s if-B. s i n ifcosif ) + ( x +2iixsinif +ii s i n if)

n ° 1 1

( a A s i n if-a A.sinif cosV' - a B . s i n if+ — cosif ) ^ o o o 1 o 1 a ^ '

. 2 . , 2 2 2 a5 \

= s i n lf ( - a u i x - a X x+x a A )+cosif (-u i -X -2uiX +iiiA x+X Ü X+X — )

^ O^ o o o o ' T v r - o o o o o a

p +sinif cosif (2iii ^ x

+2X ??c-;ia x - a X.x +a ^ x - u i B - x - X B.x-A ^ +ii i/i. +iiX A - a A x^+2ii - x^) o n o o 1 o n ^ 1 o 1 o n o o o o l a

2 *P 2 2 2 2 2

+ s i n lf (a^ - X -i-i a i - i i a X - a B.x +2iia A x)+cos if (-2ii a i - 2 i i a X -2iiiX.x

OQ "^ o o o o l ^ o o o o o 1

+ a i i § x-2X^X^x*2X^ % x - / ^ i A ^ x - X ^ A ^ x ^ ^ a X ^ ^ . ^ a / - A ^ x? )

2 c

• s i n ^ * cosV' ( - ^ x^^ ri x^-2A/a A^x+^^ Ai^a H"a X X4J^a ^ x-^* i B .

^ ^ j j 2 ^ ^ - i n 0 1 a o o o n 1

-^Vl-^oïï^ ^

• c o s * ^ sinif (2fia^ |x+2X^ \y? - ^ x^-/ia^B^x-X^B^x^^B^ ^ x^+A^ -^ x«.

-^'^1-r ^ V l

•^>o^o*^N V - f ^ o n =^)*^^> (^% I =^-2Ma^Bl^Ai;%^o)

9 2 2 2 n 2 n n 2 Z

+COS lf (-LL a - x . x - ; ^ X - 2 X . i i a x+2iia •^ x+2X. ^ x -M-A a x - X . A . x

-33-+sin If cos >f (-li a^B^-iiX^B^x+iiB^ ^ x+jiA^ ^ x)-33-+sin ^ cosif (liB^ - x-ii a^A^)

+sinif cos 1^ (-ti a A.-iiX.A.x+iiA. •^ x ) - ti^a B.sin \^

2%

Ncfw / ( c o s 0 , s i n e , s i n 6 cos0, s i n ^ 0 c o s 0 , c o s ' 0 s i n 0 , s i n ^ 0 , c o s ' 0 ,

sin'0cos0,cos 0sinO) d0 = O

271: ..271: i% 2% 2% ( s i n 0 , c o s 0)d0 = TC, / s i n 0cos 0 d0 = T- > ƒ s i n 0 > •'o J o dO = -22^ 4 T h e r e f o r e •"YS

acr

₄

a P , o 2 2

n

X - u a i - u a X - a B . x +2iia A x - 2ii a i - 2iia X ^ o ^ o o o 1 " ^ o o o o c

- 2iiiX^x+2iii ^ X - 2X^X^1 X + 2X^ ^ x - [xU^x - X^A^x -HiaQ^^x +X.,A^x*

B. B. liii.Ex

f. ^ ^ ^ 1 - N - D X 1 q "^ 1 1 2 , - A —X - l i a — ; — HX.B. 7- +n —r -^ x + —, f- ti a B. dx

o n ^ o 4 ^ i i 4 4 n ^ n 4 * ^ 0 1 '

F i n a l l y

Cys ~ • ^ I - f 3V^± + 2iiX^ +B^ (~ +ii^)- 1 liA^ I + ~2 (l^i + f '^T )

+ ~ (t^x + ^ X^) - - ^ + --j^ - ^ \ ^ _ ^. _ y . | - .

o . 1 \

^ ^ i ^ t ^ T 3 8 / j

4 0

-/l^PENDIX V

Data Used i n C(3raputing Force C o e f f i c i e n t s , Flapping imgles ond

Rotor Dei'ivative s

The values of the r o t o r d e r i v a t i v e s e t c . v/ere c a l

-c u l a t e d for a t y p i -c a l single r o t o r h e l i -c o p t e r , the Sikorsky

HNS-1 (Army I R - / L B ) . The required d a t a was obtained from

Ref, 13» The values of A and i used v/ere the a c t u a l

o

measured flight values given in this reference. These values v/ere extrapolated over the low forward speed range

(Fig, 6 ) , Tliis procedure is considered satisfactory since it is only necessary to have these values of the right order for purposes of shov/ing the effect of non-unifom induced velocity,

Other relevant data is listed below, C^ = 0,0055 a = 225 R.P.M, Y = 12.1 cr = 0,06 = ~ at 0,75 R

Blade a e r o f o i l s e c t i o n N.A.C.A. 0012 f a = 5 . 7 3 / r a d ,

Lb = 0,006

1 TIP PATH ! PLANE NO FLAPPING lAXIS /_/ ^ _{^ ^} ^ . ^ L

V

J

idU COS ^ - d D sir \ * ? 0 RCrrOR BLADE cos4-dD»in •)»ln i D c » 4 t d L ( i n 4 • ^ "o

FIG. I. ROTOR DISC AXES FIG. 2. ROTOR DISC FORCES

r

— 7 - H ^

-4-^

r

-i^^'

FIG 3 EFFECT OF CHANGES IN LONGITUDINAL AND LATERAL FLAPPING

I -o

0 - 8 0 ' 6 o 0 - 4 0 - 2 / / FUGHT MEASUREMENTS ( R ^ M 2521) !• BASED ON PROPELLOR I STRIP THEORY ( R ^ M 2521)

o Po/Pr=-x* + 2 x

0-2 a 4 0-6 0-8 1-0 OC = r/R

FIG. 4 . INDUCED VELOCITY IN HOVERING

A ' . o s c . 1-2-83'fOr o'« 0.707y& 1 - 3 8 3 ^

FIG. 5.

C T - 0 0 5 5

NOTE

THESE ARE EXPERIMENTAL VALUES OF COLLECTIVE PFTCH Ao AND ROTOR ANGLE OF ATTACK i AS GIVEN IN NAC.A. T N 1266 EXTRAPOLATED VALUES I 0 2 0 4 0 6 0 8 10 12 14 JU 16 -18 2 0 -22 -24 10 O

Ö

i

8

FIG 6

Ao,i VS p « = 121 OS 2 2 5 R.P.M.

Cr'0055

•14

FLIGHT MEASUREMENTS REF 13. MANGLER IND, VEL. DBT. REF 7 UNIFORM IND VEL DIST

— NON UNIFORM IND VEL DIST •IB -20

M

•22 •24

* »«- FUGHT MEASUREMENTS ( R E F I B ) MANGLER INDVELDIST ( R E F ? ) UNIFORM IND. VEL, DIST NON-UNIFORM IND. VEL. DIST 18 -20 JU •24

F I G 8 a^vs JU

o t

•14 •16

> X FLIGHT MEASUREMENTS (REF 13)

MANGLER IND VEL. DIST (REF 7} UNIFORM IND VEL. DIST

.. NONUNIFORM IND.VEL.DIST

•18 ^ 2 0 •22 •24

s^

. c ^

w^

(^•^. A«gï

_v<*

^c^

^' ^' >o VEU

.oqü

FIG. lO. FLAPPING COEFFICIENTS

•00004

EVOCtTf

-OOOi4

JU - 0 0 2 — 0 0 4 - • 0 0 6 •008 •OIO

•02

0 4 0 6 0 8 10 12 14

L

- - ^ ^ 1 Uk

11*1-^m^

_{^^^^^^^^^Am^s^sjsx^}

FIG

12

^vs

JU

0 0 6 •004 •002 0-iIMIFORM \ NOWJNIFORM I N n n c F H v n n r i T v •02 ' 0 4 06 OS

F I G

13

Vvs

V

•10 •14 • 0 0 0 0 4 • 0 0 0 0 2

1 _^J=

-^ ^ ^ ^ - = = ^ ^ , - - ^ -. - ^ - N( - U 1» *** -_J DN-UNIFOR NIFORM 1 •02 0 4 06 08 •10 •14

F I G

14

H^s

JÜ

NON UNIFORM INQ VEL.

UNIFORM IND.VEL

I I

F I G

15 > VS >i

- 0 0 2

F I G

16

xju

VS JÜ

•02 -04 06 08 -10 12 14 0 2 3 en - 04 •06 •08« iMEQRMJNDUCEDVaöclTY

FIG

17 V ^» ^

•02 -04 -06 ' 0 8 IO -12 -14 •OOI - • 0 0 2 NON-UNIFORM IND.VEL. UNIFORM IND.VEL.

FIG. 18.

_{Xw v< JU} JJ •02 04 -06 OS -IO -12 -W - • O O I > • 0 0 2 -FIG. 19. 'i^' '^«

w