.= ..”- :

‘- ‘-’i”’==iit~

‘““4!!!5==“i

TECHNICAL NOTES

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

/

No . 581

A. STUD~ 03’AUTOGIRO ROTOR-BLADE OSCILLATIONS

IN TFI_EPLANE OF THE ROTOR DISK

By John B. WheatleyLangley Memorial Aeronautical Laboratory

;

.- —

. ..—

r. -,GNATION4L ADVISORY COM&II!T’E!3

.

3 f176013607727 -’- --.-

FOR AERONAUT I-CS--’ ‘-

TECHNICAL NOTE NO. 581

A STUDY OF AUTOGIRO ROTOR-BLADE OSCILLATIONS

IN THE PLANE OF THE ROTOR DISK

By John B. Wheatley

SUMMARY

An analysis of the factors governing the oscillationof an autogiro rotor blade in the plane of the rotor diskshowed that the contribution of the air forces to the re-sultant motion was small and that the oscillation is es-sentially a ‘lirect effect of the rotor-blafle flanpine mo-tion. 4 comparison of calculated oscillations with thosemeasured in fli~ht 011 three different rotors disclosedthat the calculations gave satisfactory agreement with ex-periment. The calculated air forces on the rotor bladeappear to be larger than the experimental-ones, but thisDiscrepancy can be attributed to the deficiencies in thestrip analysis.

IWTRODUCTION

An auto.giro rotor blade is connected to the rotor hubin such a way as to permit two articulations, one aboutthe horizontal hinge and one about the vertical hinge.The horizontal hinge axis is perpendicular to the bladespan axis and to the rotor axis and permits the blade tooscillate freely in a plane containing the blade span axisand the rotor axis. The vertical hinge axis is parallelto and offset from the rotor axis and permits the blade tooscillate in the plane of rotation. This articulation isrequired because the forces acting on the blade in therotor disk are of an unsteady nature and experiencti hass“nown that heavy stresses and uncomfortable vibrationsarise if this articulation is not used. In addition, mo-tion about this hinge is damped by friction so that tran-sient vibrations will quickly subside.

A knowledge of the motion about the vertical hinge isof value for several reasons. The nature of the oscilla-

.—

:----

_.—. -.. .: —-.-=. ..

2. . - ‘“ ::.~~” “fJ ,A~””-—-Techn~~aimNote Nb. 581

.. . ..

tion ~ti”ugtbe known ~ri-”or”d’~rto dote”rmine Intelligently theamount of damping required by the. blade. In adfiition, averif!.ed aga”lysis of this oscillation suggests the possi-bility of subjecting all rotor characteristics to someform of mathema~~ca~..~na.lys! e~n~flens the dependenceof tho designer upon empirical rules. This paper presentsan ana,lysis of this phase of the blade motion and shows acomparison between calculated and measured results whichsupports the validity of the analysis.

,. : - -—.

AHA.LYSIS —. —.

..

The auto~iro rotor-blade o.s~~ll,ltitionsin the plane ofthe rotor d“isk will be divided in$w two parts: First, themotion about “the verti’ca~ pin. that arises from the consifl-eration”of the changing moment of Inertia of the bladesabout the axis of .ro.tqtion cause.d..by the flappi=g motion;and sg”ctind,” the forced vibration of the blades about theverticlll hinge caused by the variation of the air torquefrom a mean value of ,zero. This m“ethod of considering theblade mol,ion requi_res .g.omq.ju.stification, because it is.not obvi~us that the two oscillations mentioned a’r”ein”de-penden:j of each other.

—. __.-—

The experimei”tal - ‘-ptudy”oi ve-r~tical-hin.ge oscillationsmade with a motion-picture camera on the rotor hub estab-lished that the amplitude of the oscillation was less than0.015 radian and that the oscillation consisted mainly ofa fundamental ofthe same frequency as the rotor revolu-tions. The angular velocity of this. oscillation is conse-quently-~ess than 0.015 times the mean angular velocity ofthe rotor”; thergf~rethe peripheral velocity added by theoscillat”iori &an” safeiy be neglected, Thfs condition justi-fies the assumption that the air forces tending to oscil-late the blade may be considered Independent of this oscil-lation . In addition, the amplitudo of tho oscillation issmall enough to ju~tify the assumption that the moment ofinertia of the blade is not appreciably changed by the os-cillation; the effect of the changing moment of inertiacan , th”~e”fore; al-so be””co-nsiderod independent of the ver-tical-hinge of3’cili&tiQn.

.- ....-

,.. .+----...==—.—

-’=,

:=_——

-- .L—

—.

,..=—..=.

T-

.

.J

—

-Before the analysis i.s carried further, it is perti-nent to consider briefly the problem of hub vibrations.If theblade oscillation about the vertical pin fncluded aharmonic u“f the sa”rneorder as the number of blades, the

*.—

.-.,, < ..>L :9.-.,:. ,.- “+&“:z

N.A..C.A. Tech~ic~l Note No. 581 3

forces arising from the oscillation that acts on the hubwould add and the motion of the hub would be affected.It can be shown that the sum of the individual effects iszero for all harmonics except the one mentioned. It hasbeen experimentally verified that the third and higher or-der harmonics are negligible. In addition, experimentalobservations indicate that t-ne huh motion is steady, in-asmuch as no steady vibrations in the readings of a sensi-tive tachometer were noted. It consequently can be assumedthat the hub does not oscillate but rotates at a constantvelocity.

Let I be the instantaneous moment of inertia of arotor blade and let. 10 tie-the moment of inertia when theangle of flapping ~ (about the horizontal pin) is zero.Ihen

The assumption that sin S = ~ is justified by the smallran~e cov~refl by 13 (les& than 150). Then

I 10 (1 - sin2 ~) = 10 (1 - 92)=

Let fii he the instantaneous blade angular velocity.

law of constant angular momentum states that

I~i=K’

or

KQi’;= : (1 + p’)

o

—

(2)

The

(3)

(4)

where P’ and higher powers are neglected because p issmall. From reference 1,

@.=

a. - alcoso--blsin$ -aecos2~-b2 sin 2$ (5)

.

:.w.*S.

—4.”..

.Finally

. .- .. ,.

.—

- (,2a,oal- a1a2 - b1b2) cos W - (Z%iobl - ~lba+ blaa) sin~

...—-.

- (-a1a2 -1-blba) cos 3 W - (-alba - blaa) sin 31/J] (6)

~

.,.-

,.---- -—

.+1—.... .-. L-: ------- ---- ----- -.

- (~aobl -Cl

albz+ bla~ cos $-~ (2aoal -~aa-b=ba) sin @. . . .

.—..-. .——

—

The part of equation (’7), varyiqg with $ which ex-presses the component c1 of the angular displacement of.. —— -—the btwde oSc~lla~ion, can be more briefly written as

..-

.-

+$ U3 Cos 3$ +*V3 sin 3$

.,

(8) ““

. -:-

where . . ..,.,...-..= ;+-:. . .. .-- .....

u~ ~ 2 aobl - alb2 + bla2, VI = -2 aoal + a1a2+ blba,. . . ..,-- . .

‘a = 2 aoba - albl, etc... ..

..—

N.A. C.A. Technical Note No. 581

since ‘

The instantaneous air torque acting on a blade can bederived from equation (10-1) of reference 1

Q = {ER; P CU2 cp CL r dr - f $P c U2 8 r dr (9)

“The deficiencies in this expression arise from theassumptions upon which it is based. The more importantones are: First, that the lift coefficient is proportionalto the anglo of attack and has no maximum value; second,that the angle p, the ~~gle of the local velocity to theplane of the rotor disk, is everywhere smalI enough toequate the sine to the angle; third , an average value .8 ofthe blade profile-drag coefficient is employed which renderssornew-hatuncertain the calculation of instantaneous valuesof the profile-drag torque; four-t-h, th~ Iift torque is ex-pressed incorrectly where the tangential- velocity Slr isless than the forward velocity V; and fifth, the induced.downward velocity is assumed constant over the disk forsimplicity, which again renders the derived instantaneousvalues uncertain. The equation is, however, the only avail-able means of attacking the problem and will consequentlybe used, though with caution.

The solution Q* is obtained by substituting for U,

Cp and.

CL as in reference 1, and an expression of t-he

following type Is derived:

Q~=~Pca G2R4(Al cosw+Blsin W+Aa

+B2sin2*+A3cos 3~+~sin3$

Cos 2*

+. ● ) (lo)

The coefficients An and Bn are functions of ~, 6, A,

an , and bn and are derived in the appendix. The nota-

tion QV indicates that only the torque varying with ~

is retained, since by hypothesis “

.,~ ..... s; .-” :.. .—

~.. -:—; ----—.

—.

-.--w

6 ~,A.C.A. Technical Note No.591 —.......:=+.?-:-=-.._

The component of the angular displacement of the bladearising from Qtj is designated !a”. ?f Rv is the radl-us of the vertical pin, r the radius of the blade ele-ment dr, mx the line density of the blade, R the tipradius, and QR the restoring couple-arising frog__~a,.——then

— —-““. — - --- ——-—

QR ‘=/R” m= !22 Rv r & drRv

(11)

‘rhe angle. ~ is diagramed in figure 1, By refer-

..

ence to this figure and the law _o_$_sine.s, assuming smallangles, .- —

—.

~+’ “~; -F- :“RV (12)—— — _.-—

_—— . “3QR = f mx ti2”jr - Rv) Rv ~a dr (13)

.

.Rv ._ ..-,.: ..-.----s ~:, ~. .—

R ‘-’ “--i”;”.

It is evident that “/ mx (r - IIv) dr integratesRx, -:.-

into the mass rno”rn%n~~m= ‘of the ~lad~ about tho.- --- -----pin - then -

The equation of motion of the blade can nowwritten

.;...

Iv ‘~$+Mm n’ Rv ~2 = Q~...---- >.—....- .,

where Iv is the moment of inertia of the bladethe vertical pin. -----

:.No damping term is included b“ecause damping

.—-

y.e.rt.!w! .-4

.-. . .,+.. -—

(14)’” ‘“ :

be .

. .- -- .

(15)-. .—

,,.—---- .*,.!

. . . . . -

.—

,.. . ..-—

about.... ~.. .-..,... :of any

—“i. “. -“-r. .—e.,-—-—........ . ‘___.

- -—

N.A. C.A. Technical Note No. 581 ‘7

reasonable magnitude will have a negligible effect on thefrequency and amplitude of the steady forced vibration.

r.. —

Equation (15) is a well-known form of differentialequation, the solution of which can be obtained directlyby assuming

CS= c, cos ~+ Dl sin $ + Ca cos 2~ + Da sin 2$

+ C3 cos 3’$+ D3 sin 3* (16)

The C and D coefficients are found by substitutingfor ~z from equation (16) in equation (15) and equatingthe coefficients of trigonometric functions. It is thenfound that

cl =

Dl =

C3 =

%=

~pca R4Al—.

Mrn Rv - 10(17)

~pca R4Bl———Mm Rv - ‘lo-

&p c a R* Aa——.

~-Rv - 410

. (18)

(19) =

~pca R4Ba————Mm Rv - 410

(20)

+ p c a R’ A3.

Mm Rv - 910(21)

$pca R4B,

‘Iim Rv - 910(22)

The resultant blade displacement ! can now be ob-tained by adding equation (16) to equation (8) , neglectingall constant terms, then

t =Elcos~+I?l sin ~+I!12cos2~+F2 sin 2V

+ 23 cos 3* +’ F3 sin 3$ (23)

where

. -

8 I?, A, C;A. !J8chn5c%l ””~ote~No. 581..

. !$l =Cl+vl

FI =“DI “+ ul”-.>.

“.~a =Ca+lva.5

.. F2 =Da+lua?.: ....

.- --i= =C3+*V3.-

...F3 =D3”++ U3..‘“-l . . . . ~~,.

—..,, .- “m=

— —

.-.=

(24)

It ,is of interest to.‘consider a Simplified solutionin wh:~ch the C and J cqef~_i.i3g~.s ..a.r.q.,&eglectedand aflapp:lng curve is assumed” that has no second harmonics.

,( 2_‘. i al

$ bl=)

sin 2 $

The maximum valuo of & is, ‘neglecting the secondharmonics, which are normally quite small:

cma x = 2 a. ~ala + %Ia =-2 ~o(~max - ao)

(27)—.

(2a)

—TES”TS AliD APPARATUS.

. .._ .- -. .-. . .. ,.

The rotor-blade motions about the horizontal and ver-tical pins of several autogiros were measured by mountinga motion-picture camera on the rotor hub and photographingthe rotor blade in flight (reference 2). Data sinrultaneou9-ly obtEi,ned determined the tip-speed ratios. One rotor,the PCA-2, was also tested alone in the full-scale wind tun-nel (referenco 3) , the tests affording some necessary datashown in subsequent tables, The physical characteristicsof the three rotors tested”, “the PCA-2, $hepAA-J, and theK1l-1, ar-e given-in” tables” I, 11, arid III, res-pectively.

RESULTS

The flapping coefficients of the PCA-2 rotor are pro-

-—.-—

.—.=-.—

...—.

...

N.A. C.A. Technical Note No. 581 9

,

.

.

sented as functions of the tip-speed ratio in figure 2.Figures 3, 4, a-rid5 present comparisons -of t@ -experimentalvertical-pin oscillation in coefficient form “with va-luescalculated from the air forces alone, the angular momentumalone , and tha combinad air forces and angular momentum,respectivol~. No”third harmonics are given since both cal-culated and experimental values were fou_nd to be neg~iglble.A comparison of the vertical-pin motion during a rotor” rev-olution at a tip-speed ratio of 0..5, based on calculatedand experimental results, is shown in figure 6.

The experimental flapping coefficients of the PAA-Irotor are preson~ed in figure ‘7as functions of the tip-speed ratio. The coefficients of the vertical-pin oscilla-tion of this rotor based on experiment and upon ctil”culationfrom the angular-mome”ntum law are given in figure 8._ . .-

Figure 9 shows the experimental flapping coefficientsof ‘the KD-1 rotor as a function of tip-speed ratio. The ex-perimental coefficients of the vertical-pin oscillation ofthis rotor are compared in figure 10 with-the c-oef!ficientscalculated from the angular momentum alono.

All calculations are based upon the data in tables I,II, and III and in figures 2, 7’, and 9, which show the ex-perimental blade flapping-motion data for each rotor.. .._

DISCUSSION

In order that the desimer be able to calculate therotor-blade motion about the vertical pin, he must firstcalculate the flappine motion about the horizontal pin.This problem has been analyzed and discussed in reference1 and will not be touched upon here. ‘.

The most important deduction in this study is arrivedat from an examination of figures 3, 4, and 5, which rf&mon-strato that the blade motion about the vertical pin arisingfrom the air forces is nagligiblo compared with the motioncaused by tha angular momentum. In addition, figurafi”4~- 8,and 10 indicata that a reasonably accurate calculation ofthe blade motion can be made without considering the airforces. .-

The agreement between the calculated and experimentalblade-motion coefficients is individually excellent “as shownin figures 4, 8, and 10, except for the component Ul, thesin $ term, The poor agreeuent between this calculated

. . ...— —-.=,

.. .-—. . ....—.. -.

.—

. -.

factor and the measured on~ C:+g ~.e.eXp_lained..b.y..fi~r.e.& _.The fat:”tor 01 represents the calculated sin ~ component

., arising from the afr forces. It app eara bhat the influenceof the air forces on the motion is reflected. almost onti.re-ly in the coefficient of sin v and that the Dl term rep-resenting thins effect is of the proper sign to make U1agree more closely w~ th experiment. It is found, however,in figtlr~ 5 that ..11, which is the sum of ‘ul and Dl, dfs-agrees with. the experiment as radically as does Ule Itmust therefore be concluded that th”e calculation has over-estimated the magnitude of D1. This deduction is reason-able b~cause the actual accelerating torque on the retreat-ing side of the disk must bk less than the calculated torquebecause the strip analysis assumes .a CO,nStant lift-curveslope and does not take into account” the existence of a max-imum lift; both of these factors w“Duld cause an over-estimat-ion of the accelerating torque where the local angles ofattack are large.

The plotted blade motion in fi~ure. 6 illu8trat.~s graph-ically-the error in the motion arising from the ul term; ifthis calculated term agreed with the experiment the twocurves of figure 6 would, for all practical purposos, coin-

.- .-. .-tide..

An examination of figures 4, 8,” anfi “1O suggests that avery cl.m= approximation to the.actual motion about the ver-tical pin can be obtained by calculating the motion from theaneular momentum alone and arbitrarily addin&? to the ULterm a correction factor of the magnitude indicated in thesefigures.

.-CONCLUSIO-NS ~ ..

..1 .“The autogiro “rotor-blade oscil.lathn about the ver-

tical pin is essentially a direct effect of the rot=-bladeflapping motion.

2. The effect of the air forces on the autogiro rotor-blade oscillation about the vertical pin was shown by bothcalculation and experiment to be of second o~der, in compari-son to the effect of the blade flapping.

3. The deficiencies of the strip analysis result In theoverestimation of the air forc~s acting on the rotor blades.

r.

k,.

[

.

.-

.

Langley Memorial Aeronautical Laboratory,National Advisory Committee for Aeronautics,

Langley Field, Vs., June 23, 1936.—----- L .&

,. ... -. .: ..*E ~

N:A;C,A. Technical Note No. 581 11

APPENDIX

From reference 1, equation (10-1),

Substitute for U, q and CLas in reference 1;

integrate, then, dropping all constant terms that must sumto zero since the mean rotor angular velocity is constant,

[~= $ca&R4{(-Nao ~eoB3+~@IB 41+blPOB4+:’’B51

.

,.

,

.,.;

. ....&

;...

--–-p—..—....--KA. .C .A. ..T8.ch~.ic-al .jo:t a ‘kOo 58112

1.

2.

3.

.r-..

- ~~lhB4+&=alblB2-2>al aa33+&bl bsB3) sin 2$- ...—

( r a‘“Bl‘$4~eoB3+~e’Bj+ &asB2-~blL~OoB +3

-.—

( ‘-r ..._.+ “a 1 i “=~&~baB2+~a1~eoB ]- r+#L If’ +:+ b= -8 B3+~01 ‘4-. ~3 o

.1.;.“ .

.-.—.. . . .

7’

. =. ---: ““. .. . .. ._T ”.;—

+i~~bla & ““)

->1 aa B4}

sin 3$.

=L. . -<: :,+. . . . . . . . . . . .

-: . .,<- ---

Wh.~atley, John B. : An Aerodynamic Analyais of the‘=Antogiro Rotor with a Comparison between Calcu-lated and Experimental Restilts. T,R, No- 487,N.A, C.A,, 1934.

-.=..- - :...,_Wheat ley, Johri BG:- Wing Pressure Distribution and

Rotor-Blade Motion of an Autogiro as Determined inTlight . T.R. NOO 475, NaA. c..AX) .1933~..=. ...

Wheatley, John B. , and Hood, Manley J. : l?ull-Scale

.

——.

Wind-Tunnel Tests515, N,A. C.A,

of a PCA-21935.

Autogiro Rotor. ToRa

.-F

—. -. ,:

. . .;,-...

.

,., -.,d.. .-,

*.. ,- * , 7,-. .*

-.. . --- . . . . . -+ ------ .-.

-.. _ . .. ... ..—.

.. ..

.-.. ----:.

.—:. . .. . ... ... . .. .

a.. m. ---

,- . :-L

.-,7.

; ;--

. .

.

,.:

...

. . .

-.

—..

-.:. ___ ,.-,.:-.-->... .

-- .,?

.-.=

.. -,:,,. .

.. .. .

, . I

TABLE I. W.&2 Rotor Characteristics

Radiua Ra ft . . . . . . . . . . . . . . . . . . . . . . . . . 22.5

Moment of inertia about vertical pin 11, slug-ft? . . . . . . . 316

Radius ofverticalpin Rv, ft . . . . . . . . . . . . . . . . . .646

Bladem!iss m, slugs . . . . . . . . . . . . . . . . . . . . . 2.588

Blade chord (outer part of blade) c, ft. . . . . . . . . . . . 1.W

Lift-curve slope a

Blade

Blade

span efficiency

pitch angle 9,

(The ve.1.uea

. . . . . . . . . . . . . . . . . . . . . . 5.85

p 0.15 0.20 0.25 0.30 0.35

h 3.019!? 0.0185 0.0162 0.0114 0,0078

& .00172 .00171 .00178 .00190 .00200a

factor B . . . . . . . . . . . . . . . .

radians. . . . . . . . . . . . . . . . . -0.0157

of ~ and 8/a ue calculated from reference 3)

.00215 .0S286 .03239

0.55 O.(YI O.&i 0.70

-0.0030 -0.0041 -0. C048 -0.0051

.OWKJ .00270 .00284 .0G!99

I 1 I

N.A. C.A, Technical Note No. 581 14

TABLE II. PM-1 Rotor Characteristics

Radius R, ft. . . . . . . . . ● . . . . . . . .19

Estimated moment of inertia aboutvertical pin ll. Slug-ft? . . ● . . . . . . . 150

Estimated radius of vertical pin Rv, ft ● 0.75

Blade mass m, slugs . . ● . . . ● . . . . . ● . 1.65

TABLE 111. KD-1 Rotor Characteristics

Radius A, ft. . . . . . . . . . . . . . . . . . 20

Estimated moment of inertia aboutvertical pin II , Blug-ft.2 . . . ● . . . . . 155

IEstimated radius of vertical pin Rv, ft. . . . . 0.792

Blade mass m, slugs . . . ● . . . . . . . . .“. 1.61

.

.

.

.

N.A.C.A. Technical Note No. 581

Blad.

Fig. 1

element

RvM

Rotor axisII IF------ r “—q

Figure l.- Gecunetryof autogiro rotor blade and vertical-pinarticulation. .

.

.

.

,

.

●

N.A.C.A.~chnical Note No. 581 11’ig.2..—

— -— o? \ x’

—o

0

6

:Ahoal

5“ -$a ai—

%-l /s

c1%%

40— -

0ho -111G& / ❑

all-lm c

m.,

x3 /

,

❑

/xx’

2 /x

/ iI

aa \*+’

I—

1, u +– - ‘peed r:’io w

AA-—A I

~

I

--1-

--A

-1[

I‘A-

Figure 2.- Flapping coefficients of PCA-2 autogiro rotor asmeasured in flight.

.N.A.C.A.Technical Note No. 581 E’ig.3u

.8 0mal o 00n

Ogj o 0%

.4”~al..+o&l(i) CJX) [ c=(o)

05 xx— B ~ . .~ =— .

.+-P x x _x- ~0 xI ~ x

2-~ xA

/-.4

IE:cparim 0, n, x, +

1.6 /

. /

-D1(D)

1.2 ://’

a) !bd

/

.%

.8 “$ ,..+ I%

:c! /’o

.4 ~-’“E I&2 Im Tip.-speed ratio~ w

~❑ .2 + 3 .14I

. .5 .6

+01++,

+

J

~“ D~(+)t3 n n •1

❑—.

u. / 1’

Figure 3.- Coefficients of PCA-!2’rotor-blade motion about the verticalpin as measured in flight and as calculated from the air

forces alone.

.

●

M.A.C.A. Technical.Note No. 581

-l--- – – – ‘- – – . .L /-

Fig. ~

o VI(0)—..8

0I%

a ~.00

Ob ~3 0

..4 %a.+

o.+wwa: /

~o .:._x_x I 1“

-P -x. x x Io xI ----+--x

j

,

m-.4

erim(?ntalO,u,x, +

:4

Ti~-speec.ratic,~:UCIIh .2+ + .3 J4 ,5 6 .

.7.0 -+ ++ d= + :Ua(+:

.Q Q. _ c1 aM a n a —t c1

tI

—Figure 4.- Coefficients of PCA-2 rotor-blade motion about the vertical—

pin as-measured in flight and as calculated from-the an&l-armomentum alone.

-.

!7.A.C,

.8

d

,. 2

z-5-

0+

L. Technical itoteNo. 581 FJ~..

05

0 —El (o)

o .0/ ~ OQ..—0

i I

-x’xt-”t-x4-Jx- /P’(x)

Figure 5.- Coefficients of PCA~2 rotor-blade motion about the verticalpin as measured in flight and as calculated from the air

forces and the angular momentum.

. . . . I.

0 40

‘r— ——

——

.

\>\ \ .

-.,Pos:

80 la

II- -——Exper ment

Calcu ated —-. —-—

---+--0-1 I I I I ITT1 I I 1

I I

ive ir. direc:tion (f rot~.tion

160 200 240 280 320 3eow

Azinmth angle, $ K

I’igore 6.- PCA-2 rotor-blade motion as measured in flight and ae calculated from the angular m

momentum alone.

.

●

N.A.C.A. Technical Note No. 581 Fig. 7

.2G30

.+hwaloc)

6

a.

/

al\5 /

o\ /++

+o

4 o

x

3

x#

/+’ ,) .

2 II i! t

1 I

I ! D aa❑

u.

I

D rt cl——u—

0’ A* AA A

!1

Iba

1 ,-0

Figure 7.-

.1 .2 .3 .4 .5

Tip-speed ratio, w

Flapping coefficients of the PAA-1 autogiro rotor bladeas measured in flight,

N.A.C.A. Technical Note No. 581 Fig. 8

.8

.4

0

-. 4

o

o 00

, ,..

c1

nu n

n

/ ‘

—.”

o .‘1 .2 .3 .4 .5

Tip-speed ratio, ~

Figure !3.-Coefficients of the PAA-1 rotor-blade motion about thevertical pin as measured in flight and as-calculated

from the angular momentum alone.

&

.

*

●

L

N.A.C.A. Technical Note No. 581 Tig. 9

8

7

I I I I I I I orn \l a. I I I

.+v

:3a)~

2.+a ?)l% ---C2

—n ‘u

rl.-Ax

x.x- x

I -x.xix ‘% al

1 x

i~+” a2

--1- + Ji---+_+.+--+

0 I6 0 A A A

~—---A+ b=

0 .1 .2 .3 .4 .5

Tip-speed ratio, w

Figure 9.- Flapping coefficients of the JD-1 autogiro rotor bladeas measured in flight.

&

N.A.C.A. Technical Note lIo.581 rig. 10

,

g+-l-P

.8

.4

0

..4

.4

0

I

I I 1 n I n I I I I

A I I [ I I I I

177 ‘l”=(u) 1, ,

.80 .1 .2 .3 .4 .5Tip-speed ratio, I.L

Figure 10.- Coefficients of the IID-lrotor-blade motion about thevertical pin as measured in flight and as calculated

from the emgular momentum alone.