Effect of flow curvature due to the fuselage on the flapping motion of a helicopter rotor

I ) '~

/

/

•

TECH. NOTE NAVAL 61 T((H_{1'1_ ••.} "'~-"-: _{_ .} J' .... r -.. ·' _{._ ,.}

_[LF

Vt.,;r::~TU::::OUWi'U, 'Ci 8i~UOiHEfK (BEDFORD)

TECHNICAL NOTE No. NAVAL 61

TECH. NOTE NAVAL 61

EFFECT

OF FLOW CURVATURE DUE

TO THE FUSELAGE ON THE

FLAPPING MOTION OF A

HELICOPTER ROTOR

by

M. A. P. Willmer

JULY, 1963

THE Eet IENT IS WARNED THAT INFO"MATI co rAlt EO IN T "I'; OCCUMENT MAY BE SUIJECT TO PRIVA TEL Y -OWNED RIGHTS.

•

U.D.C. No.

629.136.35.038.423 : 533.6.048

533.6.013.422

Technical Note No. Naval

61

July,

1963

ROYAL

AIRCRAFT

ESTABLISHMENT

(BEDFORD)

EFlt"'ECT OF FLOW CURVATURE DUE TO THE FUSELAGE ON THE

FLAPPING MOTION OF A HELICOPrER ROTOR

by

M. A. P. Willmer

The effect of the presence of the fuselage on the flapping motion of the rotor of a helicopter has been investigated theoretically. This has been aohieved by calculating the changes in the flow through the disc and hence the additional aero~o moment about the flapping hinge, using a Rankine Bolid to represent the actual fuselage. It has been found th at in the tip speed range 0 to 0·6, only the higher harmonics arè affected signifioantly and that the changes become more pronounced as the length of the fuselage is increased •

Teohnioal Note No. Naval 61

LIST OF

CONTENT~

1

INTRODUCTION

2

THE ANALYSIS

2.1

The fuselage as a Rankine solid

2.2

The aerodynamio moment on the blade induoed by tbe fuselage 2.3 The effeot on rotor flapping

2.4

The caloulation of the Ranldne solid parameters

3

DISCUSSION

OF

RESULTS

4

CONCLUSIONS

LIST OF SnffiOLS

LIST OF REFERENCES

ADVANCE DISTRIBUTION

ILLUSTRATIONS - Figs.1-18

DETACHAB:çaE ABSTRACT CARDS

LIST OF

ILLUSTRATIO~ The flow past a Ranlcine solid

The oomparison bemeen Rankine solids and the Wessex fuselage Diagram showing the coordina te sys tems

Variation of I

1 end I2 with

t

ti I

1 with

t'

at

'Ir

=

180

0

" (hje)2 with (a/t)2; O·

3

<

"

(af.e)2

~

0·65

" (h/

t )

2 wi th

(alt)

2; 0 • 70

~

(

0./

.e. ) 2

~

1· 0 B

1/ao or a1

/a

o v. tip speed ratio -A1/a

o or a1/ao v. tip speed ratio a~ao v. tip speed ratio

b!ao v. " _/

"

a~ _{ao v.} b

3

/a

o

v.

a/ao v. b/ao v. asiao v. bsiao v. a

₆

/a_o v. b 6/ao v.

"

ti tt tt tt 11 2 -~

3

3

3

4

7

8 9 9

10

11

11

Fig.

1

2

3

4

5

6A

6B

7

8

9

10

11

12

13

14

15

16

17

18

... "-•

1 INTRODUCTION

Although in the past a oonsiderable omoupt of wind tunnel testing of helicopter rotors has been performed, very little is knovm about how the rotor flapping motion will be affected in flight by the presence of the fuselage. The airflow will now be forced upwards in tbe front, downwards at the rear and be unaffected at the sides. This disturbed airstrearn can cause the flow

through the rotor to be changed and hence alter the rotor flàpping motion and resul tant vibration. The degree to which the flapping wil 1 be affected wil 1 depend upon, runongst other things, the shape and size of the fuselage as weIl as the relative.position of the fuselage to the rotor or rotors. Attention has been confined in the present investigation to the single rotor helicopter. A representation of the Wessex fuselage, has been used, followed by oaloulations on a lengthened fuselage.

In addition to distorting the main flow field, the presence of the solid body wUI modify the 10co.1 blade loading due to "ground effect" phenomena, even

though the interference effect is present on only a small part of the disk. This effect might be important in hovering but vdll decrease rapidly wi th forward flight speed. I t has not been included in the work desoribed in this note.

In order to calculate the disturbed flow, the fuselage has been replaced by the equivalent Rankine solid. These solids, which are described in Ref.1, have the great advantage that the velocities perpendicular to their longitudinal axis ean be easily determined. Their shape is also such that the main features of the fuselage are r epresented. HOYlever, like most theoreticnJ. inviseid flow solutions to praotical' problems of this type, no account is taken of the wake produced by the body. This wake, as in thc case of flow past a cylinder, will have the effect of reducing the vertical veloeities downstream of tho fuselage. For present purposes, however, this effect will be neglected, for it is not expected to change the re sul ts significantly.

The theory used to determino the blade motion, is based upon that given in Ref.2 and, as in this reference, only the first six harrnonicsof flapping have been considered. It is valid in the region >there simple aerodynarnio rotor

theory ean be applied. 2

!!iE

ANALYSIS

2.1 The fusel~~s a Rél.P.!cine solli

The determination of the flow around a three dimensional bo~y is, in • general, very difficul t. Howevor, the problem is greatly simplified when the

body is a solid of revolution moving in the direction of the axis of revolution.

In this casè, thore can be dofinod a Stokes' strearn function, which is analogous

te the strearn function of two dirnensional hydrodynarnic theory.

In Rof.1, it has been shown that whon a source alld an equal sink are combined wi th a uniform strcrun in the negati ve direction of the x-axis t see

Fig.1, the stream function corresponding to the flow pass a closed solid of revolution of oval section is easily determined. Suoh a body is oalled a Rankine solid. It is solids of this type that this note uses to represent the

Technical Note No. Naval 61

fuselage of a helicopter. In Fig.2 is shown a typical Rankine Bolid super-imposed on the fuselage of the Wessex. It oan be seen that a ve~ fair

approximation to the fuselage shape is obtained. Also, for comparison, Fig.2 gives a Rankine solid corresponding to a lengthened Wessex fuselage where the length of' the body is taleen to be equal to the diameter of tbe blades.

From Ref.1, it has been shown that the stream function for flow past a Rankine solid is given by

1 2 2 ( )

1!r

=

"2 Vr sin 0 + m cos 6

2 - cos 61 • (1 )

The diViding streamline given by

'*

=

0, generates, by rotation about' the x axis, the dividing stream surface. Thus, if the semi-length of the oval is tand the semi-height h, see Fig.1, it oan be ahown that these are related to tbe strength and posi tion of the souroe and sink by the equations

(t

2 _

a 2)2

₌

_th2 _Ja2 _{+ h}2 ₍₂₎

b ₌

J~

₌

lt2~h

(3)

.J 2 a

t

From equation (2) can be found tbe distanoe from the origin of the souree and sink for a solid of given dimensions. These would be obtained from the lengtb and height of tbe fuselage under oonsideration. Having found 'a' the souroe and sink strengtb oan be oaloulated from equation

(3).

The details of how these oaloulations oan be performed will be left until later in the note. 2.2 The aerodynamic ~ent on the blade induced b.Y_the fllselage

From equation (1) i t is possible te obtain the velooi ties due to tbe presenoe of the fuselage in both x and the Zl directions. However, these

velooi ties will be the same in any plane whioh oontains tbe x axis. It is only neoessary, therefore, to consider a typioal plane at an angle with the vertical in order to oaloulate the fuselage induoed aerodynamio moment on the blades. In Fig.3, the rotor, which is assumed to be horizontal, lies in tbe plane

AFBO'.

It is at a height k from the oentre of ths oval representing the fuselage. The point P is a typioal point on tbe blade at spanwise station t at azimuth position

I/r.

Thus, it oan be aeen that it is neoessa~ to deteroine prinoipally the induoed velooity at P in the z direction.

It can be shown that the velooities at a point (x, Zl) in the plane

x

°

z' are given by v

=

x 1

2.l

_

- zr

azt v' z ~4-1

ila

=

Zi

azl •

..

Thus at the point P the induced velocity in the z direetion ean be shown to be

v

z

V

In order te ealculate the aerodynamie moment, i t is necessary to express v in _z terms of the rotor eoordinates t and~. Hence,

•••••• (6) From referenee 2, it ean be shovm that this indueed veloci~ increases the

aero~namie moment on the blade by

where

B

boM =

~

pao

ClR4

J

~~

s (s + fl sin

~)

ds

o

s

=

tR • (8)

For convenience this increment of aerodynamie moment is written in the form

boM _I 1 + I2

(9)

I

päco~4

=

where _{B 2}

J

v s ds I 1 z

_{( 10)}

=

OR 0 v 0: s ds !-L sin

~

• (11 )

Technical Note No. Naval 61

From equations

(6),

(10) and (11) it can be shown that

where H. bI

~t

=

2 and 1 1

=

F1(1jr) - F1 (1jr

+7t)

1 2

=

F 2 ( 1jr) + F 2 (", +

7t)

[ 2 2 2 2 + _~_1~_~_ al cos 2~~Ç2s .tLl!.1 +kl ). kt 2 t 2 . 2",

J

2 2 + a s~n 'f al + kl + 1 + 2a l cos 1jr - al oos 1jr

j

kl ~ . + •••••• •••••• al =

aIR

bi

=

b/R • ( 12) ( 15)

Using a typioal value o~ kt based onthe ratios o~ height of Wessex rotor above the ~uselage to rotor radius, and taking al

=

1, i.e. the semi-distanoe between the sourees equal to tbe rotor radius, the two oomponents

o~ 1

1 and 12 are plotted, against 1jr in Fig.4. It can be seen that the

~unotion 1

1 i.5 the main oontribution to 611 and has maximum and minimum values when the blade lies in the direotion o~ motion of the helicopter. This is as one would expeot from physical oonsiderations, the maximum ooourring in the

~ore position when the ~low is foroed upwards by the presenoe of the ~use­

lage and the minimum ooourring in the aft position as the flow theoretioally bends downwards following the curve o~ the body.

-Fig.5 investigates these maximum and minimum values more closely and it een be seen tha t the ratio of the semi-fuselage length to the rotor radius is

import~t. From the graph, I

1 is a maximum when

et

=

e/R

is just Ie ss than unity and falls away rapidly on either side. This feature is again not .

unexpeoted.

2.3 The effect on rotor flappi~

In Ref.2, the equations oonnecting the ooefficients of rotor flapping are given when no fuselage effect is present. The flapping of the rotor is assumed to be expressed in the form

N

=

a - \ ' (a cos n 1jr + b sin n 1jr)

o ~ n n

n=1 when N harmonies are considered.

The equations which are obtained from equating the aerodynamie moment and the inertia moment about the flapping hinge, reduce to a set of simultaneous linear equations for the flapping eoefficients a , a , and b. The introduction

o n n

of the fuselage effect increases the aerodynamic moment so that the moment equation ean be written in the form

M -M

= ...

611

nF IN

where MnF

=

aero~amie moment with no fuselage present MIN = inertia moment

6M

=

addi tional aerodynamie momen t duo to the fuselage effee t

( 18)

Now the addi tional aerodynamie moment 6M gi ven byequa tions (9) - (15), ean be m-i tten in the form

where

A.

₌

1 J 7t B.

₌

-

1 J 7t A

=

....<2 + 2 21t 00 (A oos n 1jr + B sin n

1jr)

n n

J

(I1 + I2) 00 s j 1jr d1jr (j

=

0,1,2 •••• ) 0 27t

f

(I 1 + I2) sin j 1jr d'lr (j

=

1,2,3 •••• ) (19) (20) (21 )

Technical Note No. Naval 61

and where the funotions 1

1 and 12 are given by equations (12) to (15). In this way, a set of linear simultaneous equations can be established whieh are similar to those of Ref.2. It will be notieed from equation (18) that the only change ooeurs on the right hand sides of the nev{ equations, which will now have the appropriate term of equations (19) to (21), instead of zero value which applies when thc fuselage effect is not present.

The integr&ls of equations (20) and (21) are ver,y diffieult to determine analytioally so in the oases oonsidered the integrations were perf'ormed

numerioally.

2.4 The oaloulation of the Rankine _;>olid p~~:ter~

Before the eomputation of the effect of a fuselage on the flapping motion of i ts rotors ean be performed, i t is neoessar,y to oaloulate tho

parameters al and bi from thc fuselage dimensions. Three parameters eorrespond to the posi tion and strength of the sink and source which determine the shape of the Rankine solid which will be used to represent the fuselage. Thus if it is assumed that the t and h in cquations (2) and

(3)

oan be considered kno,1U, the next step is to determine a and b. Thus from equation (16) the non-dimensional forms at and b t oan be oaloula ted. In gene ral, i t is a difficul t

problem to solve equation (2) for a in tcrms of t and h analytioally. However, by oonsidering the dependent variabIe to be (~t)2 and the independent variabIe to be

(a/t)2

cquation (2) reduoes to a oubio. Thus the solutions are given by

e

or

=

1

cos -1

[1 -

a

LL::.l~~]::J

3 ~

(alt)

=

1 (ajt)2

("2 cosh

ë

-1)

3

8 -(22)

(25)

These solutions are plotted in Figs.6A and B where for a given value of (h/e)2 the corresponding value of (a/t)2 can be svdftly found. The value of b and hence bI can then be obtained from equations

(3)

and (16).

3

ll§2..U.§SION OF jg:SULt~

Results of the theory are given in Figs.7 to 18 where the variation of the flapping coefficients wi th tip speed ratio is shovm. Three cases have been considered viz. no fuselage present, the Wessex fuselage represented by

the equivalent Rankine solid and asolid wi th the same height as thc Wessex but wi th the same length as the rotor diameter.

General1y, the results show no major departures from the behaviour of an isolated rotor. When the solids representing the fuselages are introduced, the lower hannonic flapping coefficients are not affected. The influence of the shorter fuselage is first felt in the b

3/ao tenn, see Fig.12, but the changes are only very slight. Largor effects appear in the 5th and 6th harmonies, seo Figs.15 to 18. From these figures it ean be seen that the flapping is decreased by the introduction of the fuselage and in some cases

the sign is reversed. In thc case of the larger fuselage, Figs.15, 17 and 18 show that a maximum noga tive value is obtained for arj'a

o' a6/ao' b6/no when the tip speed ratio lies be~foen 0·35 and 0-5. Indeed, it ean be seen that at

~

=

0-6 the numerical value of the eoefficicnt (arj'a

o) is less than its vnlue

at~=0-1.

For the above cases only one value of k has been used. The seleeted value corresponds to a rotor 2 ft above the Wessex fuselage.

4 CONCLUSIONS

4.1 A theory has been developed to determine the effect of the curved flow field due to the presence of a helicopter fuselage on the harmonies of the flapping of the rotor.

4.2 It has been shovm that the lower harmonies of the rotor flapping are not affeeted either by the presence of a fuselage of thc Wessex type or by n long fuselage of the same length as the rotor diameter.

4.3. The fuselage affects thc higher harmonies of flapping but not to aQY major extent. Both fuselages considered decreased the values for the fifth and sixth harmonie terms and sometimes the sign was reversed. In the case evaluated, the terms a /a , a

6/a and b!la had maximum negative values when the long fuselage was

dongidered~

'For thege maximum values, the tip speed ratio was in the range 0"35 to 0·5.

a·

-

a a , a , b 0 n Ao' A • n B b B c h k t m MnF MIN LlM N r, 6 R t V v ,

x

v , z v z

x,

z' z

"

n n

Technical Note No. Naval 61

LIST OF

SYMBOL~

= position of source and Bink

=

lift curve slope

=

flapping coefficients

=

harmonie components of aerodynamic moment due to the presence of the fuselage

= sink and source parameter

~

=

tip loss factor

=

ehord

=

semi-height of Rankine solid

=

height of rotor above centre of fuselage

=

semi length of Rankine solid

=

strength of sink and Bouree

=

lift moment about flapping hinge when no fuselage is present

=

inertia moment about flapping hinge when no fuselage is present

=

inorement of lift moment about flapping hinge due to presence of fuselage

=

number of harmonies eonsidcred

=

coordinates of point P in

OXZ'

plane

=

blade radius

=

spanwise rotor coordinate

=

forward velooi ty

=

velooities in x and

z'

directions respectively

=

.

velocity of flow in z direction

=

ooordinates of a typieal point P

=

eoordinate perpendicular to direction of flight

=

flapping angle

-LIST OF SYMBOLS (Contd.)

n

=

speed of rotation p

=

density of air

~

=

angle between OXZ' and OXZ planes

1jI

=

blade azimuth posi tion measured from downward posi tion in direction of rotation

o

=

defined by equations(23)and (25)

~

=

tip speed ratio

W

=

stream funotion Author

-1 Milne-Thomson, L.M. 2 Stewart,

w.

ATTACHED: Figs.1-18 LIST 0E-RE]ERENCES !itle, ,iltc. Theoretical hydrodynamics.

Macmill an and. Co. L td. 1 949.

Higher harmonies of flapping on ths helicopter

rotor.

RAE Report Aero 2459.

Drgs. NA. 5158-5176

Detaohable abstract eards ADVANCE DISTRIBUTIO~: DG-/SR(A) D(RN)A AD/H AD/AR TIL1(b) 120

RTO at Westlands

Ajc

3

FS/HR 2 DD RAE(A) Aero Dept 2 Struetures Dept 2 RAE Library 2 Author

FIG. I.

~I

I Z

p(x,z')

v"

)(

Ir o

...

o Ir I..

o

IJ Z .( J 0. I 1 I I I I I I 1 1 1 I I 1 I I I 1 / I 1 I I I 1 1

1---" _{'... _ /} I

----~I

I IJ til""

a (

_{_ J} .J IJ 0 111 If) ::J t.. IJ )( ~W ':L1Il Z 111 (W a: ~ TN. NAYAL 61.

FIG.2.

UJ C) ~ ...J UJ Cl)

=>

u..

x

UJ Cl) Cl) UJ ~ UJ

:x:

....

a

z

~ Cl)

a

-

...J

g

UJ

Z

-~

z

~

cr

~

UJ ~

~

al

Z

o

_Cl)

-a:

~ Q. ~

o

u

UJ

:x:

....

_.

C\I C)

u..

FIG. 3.

-

\

"'"

'"

'"

:0 ~ '" ~ ~

"-

...

-

1\1

~I

... I\J I\J

....

I\J ", 0 ~ "'\i 0 I\J + I I I \ \ \ \ / ' / o 0 t-I\J \ \ \ 0 _\ 1\1 / / 6 ~ 0 '!.l 0 I\J 0 (1\ \

r

I 1\1 I ..0 I TN. NAVAI.. bi

FIG. 4.

~

J:

I

-~

N t-I

'-'

H U.

0

Z

0

t-<{

0::

~

~

.

C>

u.

8

7 5 3

o

0·5 _'·0 1·5 KE:V

"I'

~ 1800 .,f'~ Q·3a

FIG. 5.

T.N. NAVAI.. 61.

FIG. 6.

(a}

0·'

FIG.6.(b)

0-08 0·05 Q-OZ 0-0\ 0-75 0 -80

T."N. NAVAl.. bI.

FIG.7.

'-0

---

,-

---.",.

....

~

V

./"

/ :

/

L

'\Jo%'0 • ,

·0

/

_~

=

la

/

/

6~

_Q.o R~SU~TS COMPUTEO WITH ~

OR WITHOUT F"'USe:LA~e:

a·'

'YQ.o

'001

0·'

0-2 0·3 0'4

).l. 0'5 0·6

,,0 0·1 0·0 I

V

0·1

~o

.1·0 ~ • 10 Re:SUL. TS COMF'UTCO WITHOUT F'USt:LAQE: WITH ANO

...

v---~

/

/

V

0',3 0 ·4

FIG. 8.

---..--

~ 0'5

T.N. NA.VAL- 61.

FIG. 9.

R~SUL. TS COMPUT~D WITH P\ND WITHOUT f"USEL.A.<:'E:.

/

V

V

.V

~ / ' 0·'

/ '

I

/ '

.V

/

/

/

jV

·0\ I I

IJ

I 0·\

o·z

0·3 0'5 0·6 0'7

FIG. 10

.

. /

. /

/ '

/ '

/ '

/

'IJ.%

-,,0 0.0

V

1f •

10 /

RESUl...TS COIVIPUTE:.O WITH _~

WITH~UT rU5E:LA~E:

-0'0\

/

/

/

/

/

/

I

Ö·QOI ·1

·z

·3

TN. NAVAL. 61.

FIG. 11.

/

'01

/

/ '

/

V

/

/

/

/

I

/

~

_ex

=1·0

/

Q.o ~

=

10

Re::SUL-TS COMPUTEO WITH f

WITHOUT F"USE:L"~E: ·001 I

I

I

/

/

/

/

/

V

'0001 0'1

o·z

0'3 0·4 0·5 /" 0·6 Q.3/

FIG. 12.

~ ~t /~ /~

L~~

/7/0/-/ L

L~

/JO

·001

/;;1/

/}

L

LI

/

/ /

,I' / /

I IJ

I Ij

l,fL

/

/){

_'%.0

_-1·0

I

II

t •

10

o

No

F"USE:L..AC.e::: E:F"F"E:CT.

A WE:SSf:X TYPE: F"USE:L..A~e:::..

X L..E:Nc:qTHE:NCO WE:SSE:X f"'"USe:LAC.e::: • ·0001 G:.

I

I

I

I

I1

co)

-o

FIG. 13.

-0-01 h

/ /

~

~

~

.~Y

L

/

/

C%.o)

~O-OOI

/,~

///

///

///

//I

//I

1$

/1

U:

%

_CL

.',0

o

//I

't •

10

0NO ~USELA~E ~rrECT

A WE:SSe::X TYPe: rUSe:LAGE:.

XL~N~T~e:N~O W~SSEX rUSe::LA~E.

~·OOOI //1

1/11

0//

J)fJ

I

/>5-I

I

0-3 0'4 0'5

ft

0-6

0.+/

FIG. 14.

O·003~---~---~r---'---~---~---~ ~

_;:0:0

:a I· 0 • lIJ

o

NO

~USELA~E ~~rECT

A WESSe:X. TYPE F"'U5ELA~E:.

)( LENC1THENED We:SSe:::.X F""U5e:LA~e: . o·ooe t---t---~,_---_y_---__+---+_--_H'---_f O·oo,r---t---~r_---;_---__+----7+--+_---_f

o

L-______ 6-____ -=~~=_ __ _ L _ _ _ _ _ _ ~ _ _ _ _ _ _ ~ _ _ _ _ _ _ ~ 0"

o

·

z

O· .3 0 · 4

T.N. NAVAL&I.

FIG. 15.

O·0006~---~---~---~---~---'---'

0·0005 ~---~---~~---~---~---+---~

o

NO F'U5e:LA~E: E:F'"F"ECT.

8 WE:S5e:X TYPE: F"USe:L,,~e:.

X Le:N~THCNE:O WE:S5E:.X O·0004~ ____ ~_U~5~e:~L_A_~~~~.~~ ______ ~ ______ ~---~--~~~

Uo/

_1.0 7a.o ~

=

10 ~0003~---~---~~---~~----~---~~~--~ O·OOO~r---r---~~---~---~---~~---~ 0·0001 r---r---_4r_---4_---~_r--r_--~---~

o

~=~~;:::===

0·3 0-4 0-5

fo

0·6 O·OOOIr---r~----_4r_---4_---~---~--~--~

O.OOoZr---r---4r---~--~x-~---~x-~---~x

O'0003~' ---~---~---~---~ ______ ~ ______ ~

FIG. 16.

o·ooos~---~---~---~---,---~---o

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