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TECHNICAL NOTES---

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ADVISORY COMMITTEE FOR AERONAUTICS.

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NO* 467

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SIMPLIFIED AERODYNAMIC ANALYSIS OF THE

CYCLOGIRO ROTATING-WING SYSTEM

By John B. WheatleyZangley Memorial Aeronaut~cal Laboratory . ..... _

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WashingtonAugust 1933

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https://ntrs.nasa.gov/search.jsp?R=19930081369 2018-04-30T09:42:45+00:00Z

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NATIONAL ADVISORY COh{MITTEX FOR &ERONAUITICS

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TECHNICAL NOTE NO. 467

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SIMPLIFIED AERODYNAMIC ANALYSIS 01 THE

CYCLOGIRO ROTATING-WING SYSTEM

By John B. $/heatl&y-. . .

SUNMARY

A simplified aerodynamic theory of the cyclogiro ro-P tating wing is presented herein. In addition, examples

have been calculated showing the effect on the rotor char-acteristics of varying the design parameters of the rotor.

* A performance prediction, on the basis of the theory heredeveloped, is appended, showing the performance to be ex-pected of a machine employing this system of sustefitation.

.-.The aerodynamic principles of the cyclogiro ar~ sound; -

hovering flight, vertical climb, and a reasonable forwardspeed may be o%tained witk. a normal expenditure of power.Autorotation in a gliding descent is availa%le in the e~iint

, of a power-plant failure. ... .

INTRODUCTION

. In carrying out a program of research on the generalproblem of rotating-wing systems, the National AdvisoryCommittee for Aeronautics has undertaken a study of a power-driven rotor vhich, because of the cycloidal path descriliGdby the %ladee, is called the Icyclogiro.il This rotor pos-sesses..promise in the field of hovering flight and verticalascent, in addition to a reasonable speed in level flight.

Ior the purpose of investigating the basic merit ofits aerodynamic principles, a simplified aerodynamic theoryof the cyclogiro has been developed and is here presented.In its present form the theory must be considered as givingonly qualitative results because of the simplifying assump-tions found necessary in its derivation. I?or illustrativepurpos6~, the theory has been applied to a rotor of assumedproportions, and the variat$oq of the rotor characteristicswith changes in the ~aqic rotor parameters is Sho-wtifn theform of horsepower-required curves. -..-

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N.A. C.A. Technicak Note No. 467

DESCRIPTION

The cyclogiro rotating-wing eystem 2s shown in figure1. The rotor consists of several blades rotating about ahorizontal axis perpendicular to the direction of normalflight. The angle of the individual blades to the tangentof the circle of the blade~s path is varied by a double-cam arrangement so designed that the periodic oscillationof the blades shout their span axis may be changed both inamplitude and in phase angle (fig. 2). The net force onthe rotor may thus be varied In magnitude and direction bymovements of the cams.

THEORY

Referring to ftgure 2, velocitie8 are those of therotor with respect to the air and are positive upward andin the direction of normal flight, also in the directionof rotation and radially outward. Yorces are po~ltive inthe same direction as velocitlee except that tangentialforces are positive when resistfng rotation. Blade anglesare positive when the blade trailing edge is closer thanthe leading edge to the rotor axis.

Force and power coefficients wI1l be based on the tipspeed and projected area of the rotors and are defined bythe following equations:

z.Cz = P~R3s

where Cz ie the vertical force coefficient J,.i

z is the vertical force, lb.is the air density, slugB per cu.ft.

[.

h is the rotor angular velocity, rad. per -sec.R is the rotor radius, ft,

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s is the blade length parallel to the rotoraxis, ft.

where Cxis the horizontal force t?oefficient

x is the horizontal force, lb.

(2)

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F

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... .N.A. C.A. Technical ~ote No. 467

where CP is the coefficient of power required

P is the power required, ft.-lb. per sec.

It is assumed that the horizontal and ~ertical in-duced velocities v= and due to the generation ofthe forces X and Z are

zconstant in magnitude through-

out the rotor cylinder and, by analogy with propellertheory, are onehalf the velocity fncrernent- imparted tothe air in producing X and Z.. Then if V is the trans-lational velocity of the rotor and 9 the flight-pathangle referred to the horizontal forward, let

@R = v Cos G + Vx (4)

AC2R = Vsin Oi-vz (5)

and vi = QR( k2 + W2 )! (6)-.-.

where Tf is the resultarit ve~oc~~y.

These velocities are diagramed in figure 3.

It is further assumed that the area from which theinduced velocity v= is calculated is that of a circle ofdiameter s, in all cases where V cos 0 exceeds an arbi-t-ratiyfraction - say 0.1 - of the tip speed; when v Cos e< O.lQR, the area is assumed to be that of a rectangleof length s and width 2R. This is equivalent to assum-ing that the rotor in forward flight acts like a monoplanein so far as the induced velocity Vti is concerned. Thearbitrary speed O.lQR has no great significance, sincethe extreme conditions of hovering flight and maxim-im for-ward speed are unaffected by it; its application to 13uh-sequqnt examples has been the termination of the horse-power-required curves before the velocity of, translationreach~d zero. The use of the cross-sectional area 2Rs in ,hovering flight is equivalent to assuming that the rotoracts like a propeller. It is apparant that any error in-troduced by these assumptions will be quantitative, andwill not affect the qualitative relations obtained in thesubsequent expressions. The induced velocity x i8 as-simed to be derived from the area 2Rs in all cases.

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N.A,C.A. Technical Note No. 467

2Z 2Cz flR2v= . =z ~71~~2

Ils(v+ua)$

x c~~Rx= 4pVlRs = ~w~

Stibstituting the values for x and Vztions (4) and (5), they become

V sin 8 2CZRA= -.flr +

Ils(A2+ W2) 4

v Cos e + Cxw = -fi

4( A2+ ~2 )$.

(7)

(8)

in equa-

(9).

(10)

Equations (9) and (10) may be used to ~olve for has a function of y if it is assumed that

;~~+~+ ~

is negligible in comparison with ~ QR

Dividing (9) by (10),

A 2CZR. = tan &+ -- (11)M Tr~s(A3 + ~=)+

Let the tangential and radial velocities at the bladebe designated t and Ur, respectively; let the result-ant velocity be designated U, making the angle @with the tangent .to the path circ~e. Then, referring tofigure 4,

t=QR- u~R sin~+ h~~R cos ~ (ii)

where $ is the blade-position angle referred to the hori-zontal forward

u= = U~R cos W+ A~R sin ~ (13)

u= = *t tan @ (24)

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N.A.C.A, Technical Note l?o~ 467

It will be assumed that @ is small, so thattan @ =@, Then

U*=U

u= = u@

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(15)

(16)

The angle-of-attack variation of the blades is asinusoidal function of ~, arid the instantaneous geo-metrical angle u can be expressed as ;

a = CL. + ctA Cos( v-c) (17)

where (XO is a constant angle setting with resFect tothe tangent of the blade-path circle

ctA is the amplitude of the varyiag angle

is the value of $ at which \he varying part..

cof the angle is a positive maximum

The aerodynamic angle of attackdifference betwa-en u

Then ~ iS f3qUal to the -and @ .

aT =CL-@= Cl_ Q+a-qcos(wd. (18)o

For the straight-line portion of the lift curve, ill%blade lift coefficient may be writtenCL .

CL= a~T (19)

dCL

where a i.s for the blade prof5.le, a being in ra-,. dadi~an measure. From this relation and equation (18), theinstant~neous lift L and drag D of a blade may bewritten

L =$pUeSaaT

~D=2P U2 s CDO .

where s is the blade area

(20).

CDO is the average profile-drag coefficient ofthe blade section.

L

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6 N.A.C.A. Technical Note No. 467

It is possible to express the- profile drag of an air-foil section as a function of the minimum drag and thelift coefficient, but the inaccuracies incident to the as-sumptions regarding the inflow and the angle @ do notjustify such refinement. A value will be arbitrarily afi- ,eigned to CDO , 50 percent greater than the minimum drag

of the section, when applied in the equatlons~ and will beassumed constant over the range of useful angles of.attack.

UF tO this point,---equations ban been derived for theforces on only one blade. In su@ming up the net force dueto several blades, it wtll be assumed that Interferencebetween blades may be neglected.

The resolution of tho.,

L and D forces into X andZ components (see fig. 4). results in the equations ..-

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where b is the number of blades.

Substituting from equations (12), (13), (15), (16),(18), and (20), integrating, and simplifying,

(24)

cs -- then

(25)

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N.A. C.A. Technical Note No. 467- 7

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(27)_?.

Energy is dissipated in the rotor only in the main-tenance of the X and Z forces and in the losses aris-ing from the ~lade profile drag. From equations (23) and(24) ft is seen that X and Z involve the profile-dragcoefficient; the power required for this part of the Xand Z forces introduces the same factor twice into thepower equation when the integral for the profile lossesis eet up. Adding a term to cancelterm in the X and Z forces, thet