MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL REPORTS AND ~ MEMORANDA

,The Forward Take-off of a Helicopter

By ?.

V. O'HARA, M.A,

C~'own Copyri#t Reserved

LONDON" HER MAJESTY'S STATIONERY OFFICE

1957

FIVE SHILLINGS NET

R. & M. No. 2937 (16,025)

A.R.C. Technical Report

~ - • . . . . . " . , - ~ , . . . . . . . . . " , . , .2 ~ ~ :

The Forward Take-off of a Helicopter By

F. O 'HARA, M.A.

COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR),

MINISTRY OF SUPPLY

t

Reports and Memoranda 37o. 2937

June, I953

Summary.~A theoretical analysis is given of the accelerated motion of a single-rotor helicopter for estimation of the forward take-off performance. The motion is considered in stages during which either the disc attitude to the horizontal or the flight speed is constant. Equations are derived for the motion along and normal to the flight path and solutions are given assuming constant mean values for the aerodynamic forces on the rotor and fuselage. The equations of motion for constant disc attitude have a simple solution for motion from rest (and for special initial con- ditions) giving a straight flight path, and a general solution giving a curved flight path. The performance at constant speed is considered for a general climb away case and also for climb away approaching steady flight conditions with the thrust approximately equal to the aircraft weight. For application of the theory, charts of the various solutions are given covering a representative range of the variables.

1. Introduction.--A helicopter should ideally be capable of ascending vertically from the ground in all conditions ; in practice, however, it is often necessary to gain speed, normally while airborne within the ground cushiori, before climbing away, either because of inadequate vertical cl imb performance or to provide a greater degree of safety in the event of power failure.

A theory of the accelerated motion of a single-rotor helicopter is developed in this report for use in the forward take-off case. The assumptions about • the helicopter and rotor are the same as made by Squire in R. & M. 17301 for analysis of the steady climb performance. A main point of difference from the steady flight analysis, however, is that Whereas in the latter it may be assumed that the rotor thrust T is approximately equal to the helicopter weight W, in the accelerated motion it is necessary to include the possibility that T > W.

In practice, even if no control adjustments are made, T will vary to some extent during the take-off, owing to speed and rotor incidence changes and variations of ground effect with height and speed. In the analysis, however, it is assumed that the take-off is considered in stages during which ft is sufficiently accurate to use a constant mean value f o r t and also for the other aero- dynamic forces acting on the helicopter ; in addition, during each stage, either the disc at t i tude to the horizontal, 0~, or the flight speed, V, is assumed constant.

A list of symbols is given at the end of the repor t.

' * A.A,E.E. Report Res/276, received 3rd July, 1953.

2. Forces on the H e l i c @ t e r . - - T h e motion is considered in general with the helicopter airborne and it is assumed that it is flying with velocity V at a flight-path angle y to the horizontal. The rotor thrust T acts normal to the disc which is at an angle ~ to the horizontal ; the transverse force H is parallel to the disc. The other forces on the helicopter are the weight W, and the body drag D which is assumed to act along the direction of flight. The force system is illustrated in Fig. 1.

For estimation of the aerodynamic forces it is assumed that the formulae for steady conditions can be used. Thus for given speed and power conditions (including a specified rotor speed) the thrust may be determined (not taking ground effect into consideration) at all except very low speeds, from the momentum thrust equation and the rotor work equation (Ref. 2)

T = 2zpoe2R~vf(V~ ~ 4- v~ 2 4- 2V~v~ sin i) ~/2 . . . . . . . . (1)

where eR is the effective rotor radius, allowing for tip losses,

v~ is the equivalent induced flow velocity (v~/~),

i is the rotor incidence to the flight path, and

where E P - - PR = Tu~ = T(vl + V~ sin i) . . (2)

element theory formula,

T. = {ooabc .Qi~R3(O \ 2 . . . . . . . . . . . . . (3)

At very low speeds the momentum theory is inaccurate and the empirical charts in Ref. 3 may be used ; the thrust is there given as a function of the speed, power and disc incidence in a form similar to Fig. 2. The empirical curve of rotor coefficients for vertical flight also in Ref. 3 can be used for estimating the static thrust in motion from rest. The helicopter starts moving in the direction of the resultant force, so that the initial value of the flight path angle, y0, is given by

T cos a -- W tan y0 = T sin 0c

The force H is small and in most cases at low speed.can be neglected. Where necessary it can be estimated with sufficient accuracy from the approximate formula

H = }poCDbcR~?~RV~ cos i.

if Do is the body drag at 100 ft/sec at sea level

then

D = D o \ 1 0 0 2 '

The above method of estimating the forces can be Used throughout the motion if the analys is is made on a differential step-by-step basis in which V and ), are determined along the flight path,

: .2

E P is the effective power at the rotor

PR = lpoCDbcR~3R~(1 + #2).

For the range of speeds arising in take-off it is normally possible to neglect #~ in PR or to use an approximate mean value.

To simplify the application of (1) and (2) a chart derived from these equations is given in Fig. 2, with T/2~poe~R~V~ ~ as a function of ( E P . v G - - PR) / 2~poe~R~V~3 and i ; the value of i at a point of the take-off path is (~ + y). The blade pitch 0 for the thrust T follows from the blade

Because of the numerical complexity this is not convenient for general use, however, and the analysis in following paragraphs is developed for finite intervals, either at constant disc att i tude c~,, dr at constant speed V,,, mean Values being taken for the aerodynamic forces Tin, H,, and D,,~ ; methods of evaluating the mean forces are now outlined.

The mean forces during a stage of the take-off with constant disc att i tude are considered first for the special case of a straight flight path (section 3.1.1). For a straight path the resultant force normal to the path must be zero, thus

T c o s i + H s i n i - - W c o s T = 0 .

With H = H0 cos i; this may be written in the fo rm

T W cos (i -- 0~,,,) -- Ho cos i sin i _ . . . . . . . . . ( 4 )

2~poe~R2V~ ~ 2~poe2R~V~2 cos i

This relation and the data in Fig. 2 are sufficient to determine T,, and i,,, for a given value of e,~, and the mean speed ; choice of the mean speed to find T,,~ is justified by the fact (it can be shown from Fig. 2) that, for constant disc incidence, T varies only slightly with V~. For convenience in use the data is presented in Fig. 3 with T/2~poe2R2V~ as a function of i, from Fig. 2 for various values of the power term, and from (4) for various values of W/2~poe~R2V~ ~ and c~,,,, the H0 term in (4) being negligibly small for the range of variables considered. ; T,,, and i,,~ can be found from the intersection of the curves for the appropriate power and Weight conditions. H and D are functions of V~ and the mean values H,n and D,,, are simply determined, while the blade pitch 0,,, follows from (2) and (3).

In the general case of motion at constant disc att i tude (with a curved flight path), determina- tion of T,,~ from Fig. 2 requires knowledge of the mean disc incidence, i,,, As noted above T varies only slightly with V~ but it also follows from Figs. 2 and 3 that it varies appreciably for large changes of i, in an approximately linear manner. Since c~ is constant, i,,, can be determined from the mean value of ~ and an approximate method of finding y,,, is outlined in section 3.1.2.

In motion at constant speed, the disc at t i tude is not fixed bu t must be varied in ~ way to keep the resultant force along the flight path zero ; the equation of force along the path is

T s i l l i - - H c o s i - - D - - W s i n y ~ 0 . /

This m a y be written in the form

T. Hocos ~ i + D + Ws in 2 ~ p o e 2 R 2 V ~ ~ - - 2~poe2R2V~ 2 sin i

T/2=ooe2R~V~ ~ is presented in!Fig. 4 as a function of ~ for various values of (D + W sin ),)/2=poe~R~V~2 (the/-/o term is again negligibly small), together with the power curves from Fig. 3. The mean value of ~ for determining T,, and i~ from this chart can be found directly because the solution for the motion in section 3.2 is developed in terms Of ), as a parameter.

The ground effect on the thrust at a given power in hovering flight may be found from the empirical curves in Ref. 4; there is no published da ta available on ground effect in forward flight on a helicopter but at low speed the hovering data should be sufficiently accurate. I t is necessary to make an assumption about the mean height for which the correction should be made.

3. Theory of Take-off Flight.--3.1. Motio~ with constant disc at t i tude.~Take-of f flight i s Considered in still air with reference to co-ordinate axes fixed with reference to the ground, the x-axis horizontal forwards and the y-axis vertical upwards ; distance along the flight path

3

is specified by s. With the forces outlined above, the equation of motion along the flight path is

T H D V d V s i n ( ~ . + ~ , ) - - ~ C O S . ( ~ + ~ , ) - ~ = g ds + s i n y "

The equation of motion normal to the flight path is

T H V dy w C ° S ( ~ + Y ) + w s i n ( 0 : + y ) _ g dt + c°s y"

These equations can be integrated by step-by-step methods, the forces being est imated in the way outlined in section 2. For general use, however, simpler solutions than those requiring numerical integrations are desirable, and further development of the theory is restricted to finding approximate solutions applicable to a finite stage of take-off, in the first place with the disc att i tude assumed constant.

The first equation may be written i f cos y =/= 0,

s i n ~ - - ~ c o s + c o s ~ + ~ s i n dx W dx - - g dx + dx "

The disc at t i tude is assumed to have the constant value ~i,~ and if mean values 7".,,./-/,,,, D,,, are taken for the aerodynamic forces this equation m a y be integrated giving,

sin ~,,, - - W cos ~,,, W x + cos ~,,, + ~ sin ~,,,.- 1 y = 2g J

where V0 is the velocity when x = 27 = s = 0.

For all but the steepest take-off paths it appears a reasonable approximation to take s/x 1 in the last term in the first bracket ; when the path is relatively steep, the flight speed must be low and D; , /W small, so that: even in this case the error will be small.

The equation may therefore be written

V ~ - - V 0 2 B x + A y - - 2 f . . . . . . . . . . . . . . . . (5)

where

T,,~ H;,, cos ~,,~ + ~ sin ~,,, - - 1

A----- T,,, H.~

sin c~,,, -- T~ cos ~;,i

B = 1 - D,, , /W T~I~ " H ~

f = g sin o~,,, - - - ~ cos c~,,,

4;

Simliariy for constant disc att{tude and as suming mean va iues of the {orces, the equat ion o{ mo t ion normal to the flight pa th m a y be wri t ten

V ~ d~ A - - t a n ~ , - - f dx . . . . . . . . . . . . . . . . . . . . (7)

3.1.1. S15ecial solutio~¢ of equations of motion (tan ~ = A).--I~. is evident tha t a special solution of (7) is tan >, = A = constant. This solution applies to the case where the resul tant force is along the flight pa th direction at the point considered. In particular it applies to mot ion from rest, when, with the assumption of constant mean forces, the aircraft moves off in the direction of the resul tant force. It app l iesequa l ly to backward and forward take-off.

Since the flight pa th is a s traight line

Y -- tan ~ = A. X

Hence, with (5) .

1 {v. Vo'\ x-(A2+B) 2f]

A IV Voq Y--(A2+8) 9 ] (8)

For A = 0, i.e., in horizontal flight, there is a relationship between T and e.

T H c o s a + ~ s i n c ~ = 1;

Normally (T/W)cos ~ )3 (H/W)sin ~, and so for a selected mean v a l u e Of T, cos e = W/T approximately. Hence with constant mean values for H and D,

V 2 - - V O 2

I.t is sometimes necessary for a helicopter take-off to begin with a ground run, and this type Of mot ion is also convenient ly considered here. In a ground run, the resul tant force in the vert ical direction is less than the Weight, so tha t

T cos ~ + H s i n c~ < W.

The equat ion of forward motion is

T H D ( 1 T H o~ ) V d V sin c~ - - ~ cos ~ W -- ~ . cos ~ --: ~ sin #I -- g dx

where / ' I is the coefficient of forward friction.

Hence with constant mean values for T, H, a and D,

V 2 - - V0 2 y :

I ~ H,,, (cos ~,,, -- ~Fsin ~,,,) 2g (sfn ~,,, + ~s cos ~,,,) -- D~ f~ (10)

5

8.1.2. General solution.--In the general case where tan ~ =/= A, it is convenient to put Q, = V2/2f and dQ/dx = q, so tha t by differentiation of (5), q = A tan v + B ; equation (7) becomes

(A 2 + B -- q)(A 2 + B ~ -- 2Bq + q2) = 2A~Qq dq dQ"

In tegra t ing this equat ion and subs t i tu t ing for q gives

Q S -- G(),, A, B), which is char ted in Fig. 5,

i + tan 2~ ~/A~+~//IA~+I/ G ( y , A , B ) ( (A __ tany)~ J exp ( 2 ~ - B ~ l l ) y } . . . . : . (11)

and S -- Oo V0 ~ G(v0, A, B) -- 2fG(vo, A, B )

I t is useful to note also t ha t

Q V - A , B)

S Vo

Distance along the pa th is obtained by in tegra t ing numerical ly 1/q ( = dx/dq) as a funct ion of Q/S, giving

S - -q d =~1 , A , B say.

£r(Q/S, A, B) is char ted in Fig. 6 with the zero value at the value of Q/S corresponding to ~ ---= 0 ; this value, Qo/S, can be determined from Fig. 5. Star t ing from a non-zero value of ;"0, the init ial value, Qo/S, can be found from the same figure, and then

S - - ~ , A , B - - ~ , A , B ;

also from (5)

where

~' , A, B = ~ B Y , A, B and is char ted in Fig. 7 .

I t was pointed out in section 2 tha t the mean value y,, of the flight pa th angle is required in connection wi th the est imation of the mean rotor thrus t in a speed in terval from V0 to V1 say. An approximate est imate of the angle ~'1 corresponding to V1, can be made from Fig. 5 using the values of A, B and S for the ini t ial conditions ; a first approximat ion to ),,,, follows. Fu r the r approximat ion can be made by using A, B and S for the mean conditions to re-est imate y~, from Fig. 5.

3.1.3. Solution for B = 1 . - -The general solution takes a simpler form when B = 1 ; this occurs if the body drag is negligibly small compared to the resul tant horizontal component of the rotor forces. For B = 1,

G(y, A, 1) = (A -- t an ~)2.

Also in this case d ( ~ ) has a simpie {ntegrai and

,A, 1 - - (A s + 1) + ( A s + 1) (A = + 1) - - 1

~¢ ,A, 1 - - ( X 2 + l ) A . S ( A S + l ) ( A S + l ) - - I .

These solutions for x/S and y/S are included in Fig. 6 and 7 respectively.

3.2. Climb Away at Constant Sfleed.--A take-off technique sometimes used is to accelerate horizontally near the ground up to a speed safe in the event of engine failure and then climb away at this speed. In this case mean values are taken for the aerodynamic forces as before but the disc attitude is variable to keep V constant. The equations of motion may be written

wT"~ sin (c~ + 7) --wHm cos(c~ +'~) = W + sin ~ D

T,,, H,, . V = d7 cos (~ + 7) + - ~ sm (~ + ~) - g ds

Squaring and adding these equations leads to

g d ~ C ° S y = [ ~ - - + s i n y - - c o s y ..

where RF 2 = T,,, 2 + H,,; 2.

Integrating this equation

V 2 - J ~ 'w' - J 7°'w'

@ cos 7.

.. (11)

where

with, for a > b,

I(r, a, b) =

and, for a < b,

I ( r , a, b) =

a 1 =

£ =

a 2 =

J v, W , = I(v, al, bl) +

Y b b2 a r c t a n ~ (a2 _ b2 ) 1/2 J J

I~ cos y Y b

- - a2) 1/2 {~_ tan ½y - - b + (b ~ - a2) 1/2 tl (b2 b2 log tan ½7 -- b -- (b 2 - a2):/2JJ

J(r, RF/W, D/W) is plot ted in Fig. 8 as a function of 7 for a }ange of values of Rr/W a n d D/W ; the constant has been adjusted to make J zero when 7 = 0.

The ~cheory does not give a simple integral for y and this has been found by integrat ing tan r numerical ly as a function of gx/V 2. Then

- - K 70, W ,

where

RF = tan r d ~ K r , W ,

K(7, RF/W, D/W) is p lo t ted in Fig. 9 wi th the zero values at 7 = 0 for a range of values of RflW and for D/W --O. 01.

3.3. Climb Away at Constant Speed Approaching Steady Climb Condition.--The final stage of a take-off is t he transi t ion to the s teady climb condit ion ; for s teady climb at a given power and speed V; the ordinary m o m e n t u m theory may be used to find the s teady flight values, 7 , cq, and T,. The equations of mot ion at constant speed in section 3.2 apply, and it follows from (11) t h a t

R , 2 D 2 2D W2 - - 1 + ~ + ~ - sin 7s- . . . . . . . . . . . . (12)

Also if it is assumed tha t V = Vs and T = T~ during the transi t ion to s teady climb, then as in (11)

g dxC°Sr = ~ W , - - + s i n r - - c o s T ,

and so from (12),

V? d r ( ( )}1,, 2D sin 7 , - - s in7 -- 1. g dx-- 1 +Wcos ~r

Since D/W is small and sin 7 approaches sin 7 , it is a sufficient approximation to expand in series form retaining the first-power term 0nly~ then integrating,

D . gx V v ' - L(7, - - Z(ro, w)

cot ½rs - - tan ½7 L(7, 7,) = sin 7,. r -- c o s t + cos 7, log tan ½v, -- tan ½7 ' is

p lot ted in Fig. 10 against 7 for a range of values of 7,..

Similarly Dgy w r - N ( r , 7,) - N(ro , r , ) .

( ( ) } 1 - - s i n 7 is plot ted in Fig. 11 against r for N(7, 7,) = sin r, log sin 7, -- sin 7

a range of values of 7,.

4. Application of the Theory.--For convenience in application, the methods of es t imat ing the performance for the various aspects of take-off are now briefly summarised and discussed. I t is assumed in the analysis tha t the take-off can be considered in stages in which either the disc a t t i tude to the horizontal or the flight velocity is constant ; in addition, when motion over a range of speed, flight pa th angle, or height near the ground, is being considered, constant mean values are taken for the rotor thrust T and the drag forces, H and D. These forces can be es t imated in the way outl ined

8

• in section 2. Estimations of the performance in a stage o~ take-off starts from known initial con- ditions, which include the flight path angle, y0, and the speed V0 ; if the motion is at constant speed, the disc att i tude must be varied in a way to keep the resultant force along the flight path zero.

The first type of motion considered at constant disc att i tude is the ground run, for which (T cos e + H sin ~) < W ; the ground distance to accelerate from speed V0 to V is given by equation (10). If (T cos c~ + H sin e) > W, the helicopter will become airborne ; the solution of the motion then depends on A, B and f, which can be estimated from the relations in (6). If it is found that A -- tan yo then the flight path is a straight line and the special solutions in section 3.1.1 apply; equations (8) give the distances for climbing flight and (9) the distance for horizontal flight (yo = 0). These solutions also apply to motion from rest, that is for Vo = 0.

If V0 > 0 and A =/= tan ~0, the general solutions in section 3.1.2, for motion at constant disc attitude, apply. These solutions depend on S and Q/S which are defined as follows •

V0 ~ S = 2fG(ro, A, B)

where G(y0, A, B) is the function charted in Fig. 5,

Q v and ~ -- Vo ~ G(yo, A, B) .

The horizontal and vertical distances x, y along the flight path in accelerating from speed V0 at flight path angle e0 to speed V, are given by

S__~ , A , B --.~ ,A,

t w (Q A, B ) and ~/ QQ , A, B ) are given in Figs" 6 and 7

The flight path angle at speed V can be obtained from Fig. 5. In analysing take-off performance it can normally be assumed that the type of motion represented by this solution obtains only for the limited period of transition to climb away and tha t thereafter the pilot adjusts the disc att i tude to give either a straight path or a constant speed climb, so that the simpler special solutions may be used.

, 5

For motion at constant speed, the solutions for x and y are given in terms of y, D/W and RflW, where R~ 2 = T ~ + Hh The horizontal and vertical distances from a flight path angle y0 to are given by,

- J

gy RF K yo, W W V ~ - K Y ' W ' ' "

j(~,, R~/W, D/W) is given in Fig. 8 and K(y, RF/W, D/W) in Fig. 9 for a range of values of RF/W and D/W = 0.01.

9

For climb away at constant speed approaching a steady climb condition, the steady flight values ?,, c~s and Ts may be found from steady-flight momentum theory. The 'horizontal and vertical distances in the transition, from a flight path angle y0 to an angle ~, are given by

D gx

Dgy w w - N(7 , - - N(r o,

L(V, Ys) is given in Fig. 10 and N(r, ys) in Fig. 11 for a range of values of #s.

For motion at constant disc attitude, the flight path distances are determined in terms of x/S and y/S ; S is proportional to V0 2 and so the range of, for example, y/S to cover a given change of height is greater for lower initial speed. To provide completely for estimates from very low initial speeds requires a very wide range of the variables, and for equal accuracy in all conditions several charts would be required covering progressively larger ranges of x/S and y/S. The charts given however, cover a representative range of speeds down to fairly low values only, because it is considered that the overall take-off performance can in general be adequately defined by the performance from the initial speeds covered, together with that from rest, which is given by the straight path solutions in (8) and (9) ; the charts can however be extended to deal with lower initial speeds if required. The solutions for the motion at constant speed are given in terms of gx/V ~ and gy/V 2 and again a number of charts would be required for comparable accuracy at all speeds. Normally, however, climb away is not. made at a very low constant speed and a more limited range of the variables should be adequate for most purposes.

The charts given for the different aspects of take-off are given for positive values of ~) only" and therefore supply solutions only for cases of take-off in .which there is no loss of height. The theory can also be used however for analysis of motions in which y < 0, including general decelerated landing motions. Extension of the theory to other types of helicopter, including those with multi-rotors or with auxiliary fixed wings, will be considered in connection with the general development of performance theory for these types of aircraft.

Attention is now being given to the development of reduction methods for the take-off per- formance of a single-rotor helicopter on the basis of the theory in this report.

Acknowledgements are due to W. E. Bennett who undertook much of the computation-required for the preparation of the charts.

REFERENCES

No. Author Title, etc.

1 H . B . Squire . . . . . . . . The flight of a helicopter. R. & M. No. 1730. 1936.

2 F. O 'Hara . . . . . . . . . . A method of performance reduction for helicopters. R. & M. No. 2770. October, 1947

3 A . L . Oliver . . . . . . . . The low-speed performance of a helicopter. C.P. 122. May, 1952.

4 J . K . Zbrozek . . . . . . . . Ground effect on the lifting rotor. R. & M. No. 2347. July, 1937.

10

LIST OF SYMBOLS

a l

Slope of blade lift-coefficient curve

(R~- ~ D ~ )

(~R~ 2 D ~ ) = 1

A =

TI3l HIn cos ~,,, + ~ sin e,,, -- 1

W sin c~,,~---~ cos ~,,,

Number of blades

D bl = 2~7

D RF

B = 1 - -

C

Cv

D

D ~

Do E

f

D,,JW TT;t Hj;b W sin ~,,, W -- - - COS ~t m

Rotor blade chord

H

i

P

PR

q

qo

Q

Blade

Body

Mean

Body

Ratio

profile-drag coefficient at the mean effective lift coefficient

drag

body drag

drag at 100 ff/sec at sea level

of effective power at rotor to total power

sin c~.~ -- ~ cos c~

Transverse force on rotor

Mean transverse force on rotor

Rotor disc incidence to the flight pa th

Engine power

Power required for rotor torque due to profile d rag

dO dx

Initial value of q

V ~ 2 /

11

o

R

RF

R,

$

S

T

T m

9A

V 0

V

Vo

V~ -~

Vs

X, y

W

(~11b

7

7o

7~

__~

P

(7

t2

initial value of O

Rotor radius

Resultant rotor force

Resultant rotor force in steady clinlb

Distance along the flight path

90 G(7, A, B) ' G(~, A, B) being defined in (11

Rotor thrust

Mean value of rotor thrust

Rotor thrust in steady climb

Airflow velocity normal to the rotor

Induced velocity at the rotor

W 1/~

Flight speed

Initial flight speed

Speed in steady climb

Co-ordinates relative to ground axes, x horizontal forwards, y vertical upwards

Helicopter weight

Rotor disc attitude to the horizontal, positive downwards

Mean value of ~

Value of c~ in steady climb

Flight path angle to the horizontal, positive upwards

Initial value of ~

Value of ~, in steady climb

[D/W + sin arcsin [_- RF/W

Mean blade angle

V cos i ~R

Air density

Relative air density

Rotor speed

12

,O

O

c.~

T

g J

I D , ~¢.

FI~. 1. Forces on helicopter.

/

/ EP~ - P~ '2~/'o ~a ~2vts

60 °

~ . .o -S~ °

y .

5 4 5

FIG. 2. Ro to r thrus t .

o

I..e.5

I o d m -

. . . . . d r n : ~ o

. . . . . - - 2 ~ ~

c ~ 7 ~

W z ~ v ~

i--

: O . ~ o

j . . - - - - . . . . I " - j

I

iO ~33 ~ ~ , O

FIG. 3.

~c~ FoO 7 0

R o t o r t h ru s t for s t ra igh t flight pa th .

b o 4 0 So

_ • N 4.°. ~ - ~

- - O . ~ -

FIG. 4. R o t o r th rus t for cons tant speed.

io ~o

no

I0

.i 0

40

5o

(~qmEE~] ~o

IO

"C

4 O

~ 0

B~ 1.0

N-- I.O

'6

1

,~ ~o__._..__~.~ ~ ~ B= 0.6

/ V \ V

FIG. 5.

Note.

I i Q/s = G(r, A, B).

For scale 1 # read 1 division.

14

3< (~/

I0

0

2o ~= Io0

Y ( % . A, J'O)

IO

o ~

30

2o

x IQ/

6 = 0 .6

Y (%, A. o-s) 2.o

i

0

2o

_ ~ ( ~

t O

2o B= 0-6

Y ( ~,a ,o.e)

IO

o

~4"IG. G. , ~ ( Q / S . , A , B) .

N o t e . I?or s c a l e 1 ~' r e a d 1 d i v i s i o n .

I '0

I

I'0

5

I ~'O 0 '6 = A .

\

~ / ~

. f f f ~ t . ~ F a ~ A 1'-0-__4 I 5CAI.E ~ ~.Z I" =IO

FIG. 7. ¢(qts, A, B).

6~

1.24

I"/~1' l'14 I.IO

1'14- {'I01'06 P02

FIG. 8. J(x, ~dw, D/W>.

6,o

50

4o

.~o

2o

ID = o.o~

I I,r24

0 ) 1.24 i'14 hlO 1-06 1,02

5c.a~_ FOR I" =O'~

3o

~.O. I,O

. o , 5 / ~ / IO ! . ,g ' •

25

• 3o

1"5

20 Io 5 ~s

5.5 2,5

• 2,0

55

45

(oecreeae)

FIG. 10. L(~,, 7~).

J4357 Wt.19/84] I K7 11/55 D&Co. 34/263

50 , , I.O

I o . ~ / \ I N k

~o ° % < . ~ I \ \

r 42 , ??b-- -J - ~

05o 2O

1"5 2.0 I'O

15 I 0"7 [ ~ OF_~RF_.e.~) o 5 N

i .5

?/-.% >

.4.5

Note. For scalo 1" read 1 division,

pRINTED [N GREAT IIRITAIN

16

R. & M, No. 2937

Publications of the Aeronautical Research Council

I938 Vol. I. Vol. IL

1939 Vol. I. Vol. II.

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