Propellers in yaw


Material Information

Propellers in yaw
Series Title:
Alternate Title:
NACA wartime reports
Physical Description:
60 p., 12 leaves : ill. ; 28 cm.
Ribner, H. S ( Herbert S. ), 1913-
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:


Subjects / Keywords:
Propellers, Aerial   ( lcsh )
Yawing (Aerodynamics)   ( lcsh )
Aeronautics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: It was realized as early as 1909 that a propeller in yaw develops a side force like that of a fin. In 1917, R.G. Harris expressed this force in terms of the torque coefficient for the unyawed propeller. Of several attempts to express the side force directly in terms of the shape of the blades, however, none has been completely satisfactory. An analysis that incorporates induction effects not adequately covered in previous work and that gives good agreement with experiment over a wide range of operating conditions is presented herein. The present analysis shows that the fin analogy may be extended to the form of the side-force expression and that the effective fin area may be taken as the projected side area of the propeller. The effective aspect ratio is of the order of 8 and the appropriate dynamic pressure is roughly that at the propeller disk as augmented by the inflow. The variation of the inflow velocity, for a fixed-pitch propeller, accounts for most of the variation of side force with advance-diameter ration V/nD.
Includes bibliographic references (p. 59-60).
Statement of Responsibility:
by Herbert S. Ribner.
General Note:
"Originally issued December 1943 as Advance Restricted Report 3L09."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003805565
oclc - 123958287
System ID:

Full Text

(JAfc4tL- 9^

ARR No. 3L09


December 1943 as
Advance Reetricted Report 3L09


By Herbert S. Ribner

Langley Memorial Aeronautical
Langley Field, Va.


* ,5:5


NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.

L 219

r .1 ._ ...
F I~i

Si. ;t,: .

Digitized by Ihe Internei Archive
in 2011 Wilh funding from
University ol Florida, George A. Smathers Libraries wilh support from LYRASIS and the Sloan Foundation

htlp: detas propellersinyawOOlang





By Herbert S, Ribner


It was realized as early z- 1'C9 thr.t a oro"eller in
yaw develops a side force li':e that of a fin. In 1917,
R. G. Harris expr-s::-d 'f-rce in tr. -. of the tonr-.-
coefficiert for t:.' -.:.-awed rin;..3l]er. .' several t-
tempts to ex'jress t:. asie force :-t etl. in terms of the
shape of the blde.s, :3'V \-:r, .As >en cC.npl'.tely
satisfactory. An r..lrcIls t:..:t :_..-:-.r .rates tn-.:cittion
effects not a ILe- .:at ly : v:r -,. in ';'.-vious work .ind that
giv s g-cd a trint with -'. ri .r.t1 over a wi d. rn.'r of
operating conditions is :rr-o..r.t:. 1 ..r:in. T'. pr ':cnt
analysis chi:.s t.- lin n .r. ..: : ax, .n w a to
the form cf ;t-:e -i.. .-force u:..:-s ion anr that th. icffec-
tivc fin ar-a r.e.7 b3 t-'.?-nr. .: tL.,.. Dr'- 'Ct-... 31l- arZa of
the pr n-'cller. T .c- off *c"' v --. :- ". : L-. :i 1.._
or t.! r of g :..nd t 'i n or ,pri t:' -,r.r.j,.r :. '.-. ".,
tnfra at tl.? prooul.l:, dick m: &u 'nt. by UL. in:w.
Thu variation of th.: infl..;w v: lo.?i'.', ::.r r. "y .: -r, itc'
prc pulcr, a2ccuntr for m .O3t Co' 'r. t :'n 01' C.-
force- witin- av'.nc-.dian..tar ratio 7/.D.

T..e propeller forces. ie t' an an '.-ila:r :el.'l it 31
pitcn are aico analyzer and are :i'. r e vFry c-rall
for tne pitc:-. .- v loc' l t t '-.: ac..Ct 1;- l ::: r:'e. ize
in maneuvers, witn tr- -::.:cet.ton, of t.: GTr.i..

Fur-ther con.:1cri3r,2 nre? A .ul-'ote tinrl nroreller
in yaw develops u to one-th..rzi 2o'r ":-I: .'.7rce t.. r. a
si.g.le-rot .cinr pro;-el.e. .-e. ...:e:. ".: -- t ta int
propeller experience. a t it:-i..rng n:nenr ir. _a.._d.ti n to
the side force. T'ec pitchin- moment is of t..e order of
the moment prezuced bt" n for.e eqjal to tL.e si:e force,
acting at the end cf a lev-r" 0.;'..: acual t- c. :- ropciler
radius. This cross-courlin betv.en 'itch a :. 'aw is
small, but possibly not nre litibie.


A corrocticon t to te side force for compressibility
is included.


The effect of onwer on the Ftability and control of
.rcr:ft L- t'ec.min: of greater iniportasnce with increase
in :.L--ninr 3outout and pro-nel le-1 olidi ty. A.n important
part of this effect is duie to thfe serodynamnic forces
'Yer"i ecedr b',' the nrcpoeller mnder an.y eviation from
L!.iforrr. flight .-rallel to the Tihri~t axis. The remaining
p::t is di-P to the interf.'rtnc- tcet'.":cn the propeller
linst re.i and thle othlir arts of' the airplane structure.

A nri.-ter o.f 'r.-.ers hPve :onsidcred the forces rx-
peri-nce-d ',: 1.-- -oropel 1-r. It vwas pointed out in 1909
reference 1), ars c ettly Ly Lrnch!-.?eer, that a propeller
in ycva dcvelo:- c.A '.sid-Proble side force. Thr basic
E-nalysis wv's o'tl] i _,.d :-i 1'K.12'" bh G. larris (reference 2),
who sho'-cd that 23 tIitzthin n.omrcrt -ri~~.e s a well. Glsuert
(refere-zFS -"rd 4I extr-iurjed the method to derive the
othL-r stab'lit;, der:vativ'es of oropeller.

arr~s an'd fluert e-.nprcs-ed the forces and momcnts
in terms of t. tnruct arid tort:,ue ceff cients. for tl-e
unvyred "ro-'eil:r, v'hi'i hereT prcs'tsblv to be obtained
exn, r-:rc.,tal ,. The analyses did rict take Into account
certc-in lrton rd-tion .f ts snaloknuc to tne downwash as- v'it'h a finite wing. It is notw'orth, that with
a ser 1e:iri1ail factor tnr Harris equation for side force
doe. giv:- E0odO Egrttr,.ent vith rxc .rimrrnnt (see reference 5).
Pistolss. ( rf1 rcnce 6.) it 1- 33-n.ideredi r the induction
Cflicts but his tr atmzent was restrict d to an idealized
part icu1. r carse. 'sinrmani ar.3 '" iinig freferen3i 7)
in 195 published an aianalysi? neglecting the induc'tion
effect-; the tretitm,-rt iprera-s almost identical with the
account iv.?n in Lt'955 0y GOlasuert in reference L.

There have been s-evral notaol e attcrots to express
the si d' force directly In terms of the shae- of the
bliad s'. s.arsto-v,; rreferen3-e ) presented a detailed
anrl ysi. in l0 l Lhat iegl-,eced the induction effects.
"i.sztal (rferen c 1) p'.blished an invstigati on in 1952
that did not hav: t:is limrit-aton arid that is probably
the irost accurst,- up to 'he pre.sent. ;'lsztal's result,
however, is 'na 9 er-y .orrmplex.-:form from the point of view
of both practical com.nutation and physical Interrretation;


there is, in 'addition, an inaccuracy in the omission of
the effects of the additional :;--arent-mass of the air
disturbed by the sldewash of t}:- sliostrer.

Very recently '"Tr:h, V';,:-te, -;nd qrum'an (reference 1)
published an analysis th-:t relates the side force "..rectly
to the plan form in a very es~-:le manner. -.ferenso 10,
bh.k.' .r, (1) does not include the o" 1lr:-; inflo w in the
ai-I~:.sis .:,d (2) applies unste o.:,-1' ft theory in an im-
pr-r:;.r manner to account for the i.:.muction -ffects. As
a consequence f (1), tri equations srt ly in arror at
hirh' slipstream velolties. ,s a consequence of (2),
the c-1:,ti-:-. f4l to pr 'ct he substantial increase
in ir]e force that -.- ": .nt cows is ri-id.d 'l
rat .tijn. T.-. ..' r use of un t t er con-
sistod in :' 'ii formulas thr ,- 'i t1 +. cs of a
finite riU.oil with an cs to:-:.t ally :; =;.inar wae. .
vortex o1:?.'s shea .d ..r i. fi lt ~ i. l '. 'lor' .?.
the int-'ierence flo ', s. "'!].- ibut a' *..,:, this recti-
lin: : r '." *.".:. Th: corr. .r-,i.. v)o t x ,-,: s S' a
oro.-eller blade in yw however, I ,..7 tho he2 cl
pat, i traverc d L; t'-. blade. '". 1 rerence I L.V-, is
quite different Irr the flow for the case of a, rectl-
li ~ne-r w-c-. In fct, It csn rswn that the vortex
loo1.s s-ie.a do-'r. the unsteady .'t alij themseves in
such a waj" as to or-d 'ce an inflow ,*:.t. s retry. h' s
ntl.sir-L;et is one of the two :'. ition effects that
will be d-'r. :.:d in the ,-.t.t analr s r 1 momentux
cco.n ; der-' t ions.

To tu u ", t ; --* :- j .* .. : 1 n' ? L .r ,-.n
the bl d 3 .-- t: "--.,r :uff .: .. .t Ov. th
V, c.. *1
L1 r t l- F-.. t L ;
vh l r : of -?:.:' ._r 't v. -:.' s '.. t ..
2..;alysis .-t 's t Lh-- r-,s .. i '-t rL nct Di r t --
f .ct wil lV -" f' ). .',. rE r ...t l s .. ": -- 0','; "jl' t a'L of
n ,r i s". .. i o v- ? .'t -'". t : :' ,s ". t '' *'l -' t r"

tha-t the I'-n -r.. Ic r :: -t. t : ; ..
side- i c r .. -

Th- proj.-ct -] :-d: *.. t"..: jr d co. t- ;':d; ...y t .
oe *I r o, r. r ..

p'.oj'i ctea .s e -.r i I ? v t r.
by onjl e- center lin. ,. ,7 .' p .;:i v i. t .; pr >-
j.cted by .: s:im;'!. 'u. rn i ., n.L. T*. ". r.-., center line '-i- the e;.-'.3 of .-, ti. ;.

the eff?c';ve aspect --tio is cf the order of S. This
equiv-jlent fin vrea ray, with small error, be regarded
es situated in the inflow at the propeller disk and
subject to the corrr'sonrirg augEimented dynamic pressure.
The vari action Ii th V/nD of the dynamic pressure at the
prooeller di-k, f)r a fi;::d-nttch uro-eller, therefore
.cc.ounts for riost of t-he? v'iration of side force with


The forinmulas of tie present re-ort ref'r to a system
of hod:- axes. For s igle-rotati..t. propellers, the origin
is ct the intcrsZctio.i of the E.X-j of rotation and the
plane .if rotation; f'r dual-rctatin.g propellers, the
orinin is -,n the -..:'i -.i rotcti.o.i Lelfway between the
pl..nE3 of r'ot'.tio!L o.f tre fr'-nt and rear propellers. The
X-xs iS conl irpt -* tnL xiz cf rotation and directed
forward' th n- Y-ax-.C c' 're-tced tc the r-ight and the
Z-axis i dirc-.t-. l do:-:n'~Er!. hIre synls are defined as
fol low's

D oro'rrller di'3met.r

S' lk reca trD2-,.)

S w'inr aurea

P tin ridius

r radi.u to any blarc-e 1-ment

ro rinirmumr ral! us a.t sAlh-; shnk olad- scctions
devcloo 1it iTaken as ).R)

S fra'Stion of tip r2'di'is (r/')

xo value of -y orrT- sonding to re TR

:s, ratio of rpiinn.r rad 'us to ti) re.-l'is

5 number of bladep

b bla le sztion -hord

c wing referen,: chord

r h/- -7

p. relative blade section chord --- or -
7bo R (b/)

0* = ,

a solidity at 0.7 7

V fre -s tr er, veloi t:
/ r I'
V + r' -1.
a in flow factor i : in edx ,
\ 2 /
s-- of -i in free strc

oracl acc -ler. tlon

S~,=. leration of :-al-ity

V, 1-.': l velocity t or- :il er sic +" fL + a)[
V9 vel.o i" c r t '., n L. relt on ; c e1.--.. 9
., relative vind at bil element n(Tnr slin-
S. t i -. -. i ,* :

q f--blade -- l' e -.r- :'1
.. .... "+ ._ *, '. "1

f;'al .-^ t;ar 1 + r.-

l + -:'l 3)

n r"voluti.,rs ,' : -,n .

J advar'!- e-Li i.r.-..t-r .tl.., ..'.:,:

p blade ian' to r'fref ?-",,z, ', "d



jS blade angle to zero-lift chord

9 antgl.e of blide relative to Y-axis measured in
direction of rotation

6 P-ffecti'.yF helix sanle including inflow and

rotation tan -I

Prle of yaw, radians

a -ffective angle nf attack ol blade element (o- ')

C angle of sidewssh in slipstream far behind propeller

e' no:niinnl inruced argle of sidewash at propeller

' eff- :'tive .,.. induced ngle of sidewash at
1ro 1jl ,r ci -k

v,, sidev,-sh -voci clit- far beh:'nd pro, pellr

sir,)l:.rne lift .. feff-icel. nt
z blade ecti.,n lift -zo ffic'ent

c 0 blA:- s-etion profilP-dra, cc:'c-ffici-nt

c7 elc-'e of ol.d.e section lift curve, per radien
"a (d?/dl: v.?rage value taken r,.s 93. x 2n)

dF fcr'c c1 r-',one-nt on a blade el.-ement rl direction
:f[ decr s-ing 9 (See fig. 1.)

T thrust

T th-e-sFt r!cffici-rt (T/pv Dl'

C th:rst co-ff'i -ent (/n2L)

Q torque

tQ torque f. I'fi i nt (,/pF?2D3)
w weight of air-"lane

X,Y,Z forces directed along positive directions of X-,
Y-, and Z-a;,es, respectively

L,M,e miorirint about X-, Y-, and Z-axes, rr lectively,
in sense of ril.t-.n-' i" secret; in e-:-ndix B
r.f fi ure 9, refers to the fr: =-stream "Sch

" : rfective "'"s: number for propeller side force
( See ap:: ni-] ? : )

A', B', C', 3' functions defli-r '1 in equations (4.)

a', b', c', d inte .' s de:. !- "; i.. tions (21) n
( ."-O

I~ntcrW Jcf J 1~S:

~;cde-srea 1. -,

te -,ns (1 .-' (i2)

ne .- *- , (41)

I in4 2)

in4.- ., l :.-7 ,.,- in d

d ;'. .- i by i..tion (2l) ( -- for dual-ro
p r ~.el, Ilei s)

i ...- ; at : (') (Zer for ~.: L- ro

j- n : ...r'. on )

correct on factor .:*f,.- .... *..tlon (~?I.)

3 ash fa"tor def'.-. 1 : t on (3)

s i1. n-:i .E t -: ni Z1 .

. r'3 r .-- i'. r r -
ic ." c -f *,,". *',,- s1 .. :*' ,-' L / i -

t '. in


1 j '

St,;.inj- -. n e- i :. nt ---- ,-
\ .. '" .-- / T

( aT

b1, b2






r s

C '


i-tegral def 0 ..:' by f: '.atlon t i.)


C nit.zhi-niT-woment d.eji vat-ve with respect to yaw

C.,' side-for"'e derivative vith r-spect to pitching
'1 q [ 6C.,,-
q t ,

C' ,i1 tchbiLng-rontnt derivative with resoect to

q ,5 r, "-e C
pitaT -lo
O /-- I

S lroje-stedc ride arfa -f prorl ler (See footnote 1.)

A ;.nF;r rntio

Subscri oft;:

0.7 -'- Rea-isurd :it '0.7 R station (x = .75.)

c di -i.: -' by pVj2 if a for-.e, by pVI-D if' a moment;
,d -jira..-- quantitlee corrected for compres-
si.b iit,- in -")endi' z 3 B ,,1 fiLure 9

e iffreti -

k Cid.-- 1 i-k s the o .-ala.I 1. t'. Z to desl-r.ate
a 3 r -)T o -o iPr bi ,de

Inay. '. x.: :r. .

stall .t st.ll

A bc.;r a Fymrbol denotes effective average value.

P*;' A' ': C.!TSQ

r lc,-re:ier in Steady A1xial Flight

Thr. -:-.on ,' wn? r fi., ; r is part of a right-
h9rnd pr'r-. l': r : '.. i: ,'ng t: the right and advancing
urv;arid.. .-. ,'* nc:.1- of the relative wind are V
and Vf, rre t' i the axial velocity including the
Snflo," L&n i is tr.- rotational velocity including the

slinstrerm rotation. Th- force co-1:,c;n.-it in i:-e 'direc-
tion of ef~crejsin.z is:

IF = dL sin + KD cos /

(. c s + Ao cosS
2 asin2

= V(T,""1 !. q1(! (1)

and the contribution to the thrust Is

dT = dLT, s d i

S--" bur -

= b n. r. (l)

'i2:' K-quat ons m y be '..v '* r 'Y t3 > e
the ter-,s tc nmiC -' rt:n 1 f"r. Tna'aUoh u' V, ='.'i+),
th,-rc r :sults

(I + a)2 b f

S- --
T = t- ) ilx

whe re

It- V

ad'r -.-n



Proneller under Altered Flight Conditions
rorc- comnonents on blade rlement.- Tn equations (1)
and (2) for dF and dT, V2 occurs exnlicitly in the
factor v and implicitly in r and in terms depending
on 0; V6 occurs only implicitly in 0 and in terms
-1 9
de-endin, on The relationship. is t = tanI Va
which can be seen in figure 1. By partial differentia-
tion, therefore, the inrcr.ments in dF and dT due to
anv', chanic s whatsoever n V8 and Va are, for
fixer6 blMde angle,
6(d) =_d ,I) i 6(dF) 6(dP) Add
(d) = dv + -- dV
6 ) V, I Va 6 v0 V a

nd a si.nlar e;xpres. sion for 6(dT). The substitution
of equations ( ) and (2) *.ives, .when put in nondimrensional
1 + a b 6,: 1fl f1 +'
5(1u )= d V +dV ---1+- 1

u9 Qa ''a
(1+ a)2 b F 6 6t 6 \1
D dye 6v, 64' '
The following abbreviations are helpful:

Ya 6' 6r
S 2fv 6v f

S- +.

"a 7a 6a

Va ,, 6V a a '.


where fl 'f t, are j---'ned n
1 x
resoectivsly. Equations (3) b come

(. + f) ( )
4 -H.7" e

(I + a)2 b2 /
'- --G L

' a

.l.--tions (1) ricd (2),

'- + B
V "

+ D-
+ D' --)

S (5)


Where all the factors ere '- .' .-:3 tonal.

n .t-: : --.r r. '-- -- l 11er. -
qu ..i' on ). ve tr com~onent-force I" due to
altered fii ...t con,- i'ons; on an element of a sil- leT'.. .. L.:i
.1, -:D '- fore .... momentnt increments x;. rienced
1 : the c:, -let pronller of 3 bl.' s, with r-sp-:.t to
- '. r.. ,y -'-:..s r .ia n r- 2, r !., e written as

^ ^s

zi--T r

Y = / (dF)k sin B,
1 o

k -I
z = > / ,,(,-ii cos 9
k=l 'r,




=- 7 F r-:.),., r
"T 0r


*. I

L -.)I





X T =

k=B R
S= r(dT), cos 9 (11)
k=1 r

where the subscrlirt k refers to the kth propeller
bladc. In order to obtain thp nondimensional forn X,
7, _, and T are divided by FV2D2 to give X Y, ZI ,
and c and L, ':, ;, and Q are divided by PV2D5 to
givi L," and q,. Tn the equations (o) to (11),

5(d'r) b-con-e 5(dF',),, 5(dT) becomes 6(dTc)k, and
r x
r bEcomes r- Th liitrts of integration become
D 2
x to 1, where x -
x, 9 F.

St .1_ilit- dr .i vE tives of propeller.- Th' analysis
up to this nmint has been of a gen.r-lral nature' in that the
formula. are ao'iicable, for a fixd-oitch propeller, to
an,' t:/n,- of devition fromr steady axial advance that
is, t:i- formul];s in.-y be used to calculate all the stability
derivatives of a fixed-pit.h proocller. In addition, the
form.ul:.s sr.- applicable to those 3tabilioty derivatives of
a c:n3stent-3r p.-:d nrooeller tht arc not associated with
chang.-s in olade rsnile. This restriction could be. re-
mov.ed, .ow2-vtr, by extending the analysis at the outsa-t
to include a tr'rm in d .

A particular stability' derivative can be obtained
by detcrn.ining and substituting nri equations (5) the
v-lues of 'd /'a and dV q,/Aa rnpropriate to the
mo tion under consideration. For dual-rotating propellers
equations 'i'))must be v t u;. ladepenrcntly for both propeller
sections with s.gns ':mrooriate to tne respective direc-
tins of rotation. Values of dVg/Va and dV /Va that
are a'vre-sae icr both sections are used for each section.
Note that dV9 is the in the component of the
effective relative wind acting on a blode In its plane
of rotation and dVq rust therefore Include the effect
of any changes induced by the -otion Jir the rotational
speed of the oropelleir relative to the airplane.

-"h, -e ::-_ible unacceleratei motions of a pro-i:11er
co-oprise fli:,Lt (1) at a steady .angle of :, r, (2) at a
t3 .-1., r"-:e of .itch, (3) with an a'i._ular velocity of
s-v, (Yi ) '"th ai :--L .r velocity ; of pitch, (5) with an
ca,.alar velocity of roll, '6) with an increment in for-
ward speed, a I ny combin-tton of these. It is clear
from the s,-ir-try of the oro')-ller that motions (1) and
(2) are similar rnd motions (3) and (!.) are similar.
Accor3:.ry3, of the six possible deviations of a oro-
-i ler from a riven nm n-c of steidy axial advance, only
four are distinct. These four may be taken as rnrle
of yaw -i, .ru ar velocit, of pitt' q, ,n:ilar veloc-
it of roll, asia increment in f-'r\'.'-rd velocity .

lauert has n-,:-,n in reference 3 that neither y.~'ed
fll--t nor fr -' t with an ,ly-ar velocity of pitch,
v;rtn these disturbances :- small, .'..i,:;.- the torque on
the propeieir. 3c i,:'. -ly, neither r.-de will tend to
-o" the TotationaL veloCv ty, -r'j derivatives with re-
spect to oav :o '- r viloci-' of I' tch are io '.-.:de, nt
of th rate f r~r of ;*:ein torque '.th en 'ne revolu-
tions. :'u-ther-ore, res.~ ts for th--se derivatives ob-
ta ,- -. fo a fixed-oitch onopellr are -qually :.- liable
to a constrnit-s'-ed o:i r ler be-,_.- t': const-r-t-sp, ,-d
T,..-h ri.'n is -i-)1- br ibt '.to 0 1 ration.

O:it:h .n.lar velocity of roll in,: increment in for-
varl velocity clearly afect t-r torque of the
rhe -n:, _ne vill attempt to alter its revolutions to att'ain
an el Lib.'r um value. If the pro-ll1er has fixed pitch,
the 3.'.st.Lr-.nt will t-1V place 1-f.d its amount will -.i.pend
L1"onl the law of -rlati- .1 of e. in-3: t-r. ue with en1'n
r.- vl':ti -1n for the srtic'ular enr.:..e us-.d. (See refer-
r-nee T.1 Tf th,,' h r j-.-17 ,.r is .f t ,-r cn."rtent-spee d t:.7l -,
th; "ite'-i- .s ,. e -e rni' -: vvi' l Ettet .r tto slt.-r th- tade
pitch; tno r:sultin h'at 'n i. r ,r na.ric t r'i.e o .,r ses
th: chan-e 1.1 1 \'1 ri i The fl c tuat;ons ir rt-'tional
so-'-ed an-1 th-. : J :oc1 t- v ria.i..ns in ;a -r.;:ini r.ii c torque
,nd t.-rust of th', r, l .:r : : th' n f.I-; t i l 1; r 1 -l t-2d
to t l-: 1*. of 1 -C nt Ir t h- "- 'tch- 'mn l -.- me n~-ni sm u.rd
th-. d, 'n ..,. ".- Its It rS.t]'i n, 1 ._. r -f.r. n,*: 11.)

The '.rcrient e- :Trt -.ill bc lii'; ted to a study of the
effects o-f y -",v -r.J ;.f en.:'.lsr '-el.- i t:,- o'f oitch. In the
f-ilov'n,.; secticn n :V-i," a.1 d'.' '.". are evaluated
for ya"i-? d -roticrn.

Propeller n Yaw

Fatio d \'. a for vawNed motion.- Th7 increment dVg
is the coia.lone.jt par'ill el to V9 of a side-wind velocity
co3-ruted '- foIlo,'s: The velocity 7, is regarded by
anral.ogy with vin i theory as n-aszing through the propeller
dils at 3i. anlF E c' to the axis, where '\ is the
san. of :,o'r -nd E' m.T:a be termed the "induced sidewash
-n '." i'fig. 2). The side-wind velocity, for small
%alLtes 'D both 'V and E', is accordingly Va(4 ').

7h: sidve'ash arises from the cross-vwind forces.
These for.esF *"-r- the r-ss-wind component of the thrust
T sin 'rr and of the s i- d force known n to be prniuced b;
yn Y cos 'i'. (S-e f'. 5) Thp snalysis is restricted
t3 :3al 1 $1 ; these ccirr.ore nts are then ouproxinately TV
anid Y.

If the d-s'. ah velocity far behind the propeller
is ., t he in'f. u --d s: t-.swash at the rnioprcller may be
ta',en as v by 'nsaloy :vith the relation between the
in-ducLe do' .n'"wsh it a finite win-r and the do''nv.wash far
bchipd the wing. Vote that 1 ram'.nter may be considered
"far" behind, the nropell r as re'i rds the axiel slip-
streaim "elo't;i; -. percent of tnc finril inflow velocity
is attained at this distance.

As s first aionroxi.mition, thrust and side force are
asuir:n.r- to ue unifor-rly cl'stributed o':er the propeller
dis'r; corr-'ctions due to the actual rdistrito,'tions ere
Inv. sti. sted in ao.-nd ix A. Under this assumption the
~mEniu:.. tcory, su-corted qualitatively by, vorte.: con-
sJ I* rations, shows the slitstream is deflected side-
vise : a r.l'id .ylind. T'he ~dewise mo tion induces
a flo' n-f '-.r around tl-.e sli .strcar; a3 in figure 4. The
trPns"c- rse M,'orentum -f this flow is, uccord'ng to "unk
preferencee 1:2), -lqusl to the transverse momentum of an-
other 'yicr of sir hriavln, the asmo diameter as the
sliOpstr-'a at aLl points aid moving si.dewist with the
same vylocnityt as the l ipsLre-~m boTIlundary. Dote that the
air ,r: tre sliostrr am ha-s a ;reater sidewise com-
pone-nt of '"elocity than does the slistrne9m boundary.
f-r buc l ,i' tnh ::ro.:;Eller the ratio is z = 1 + 2a.
The tim= rate of change of the transve=rse momentum of
the rir flowvin. .rt free-stresm velocity through this
sec-oni cylinder should be inmlitded in setting up the
momentum relatinr.s f'r the sidewesh.

r'/ eq-:t-ini-, the cross--"'ini force to the total time
rate ci ?fh,.i.p of momentum,

Tir + -- v + D' -

to the first -rl-r in x where the first term on the
r, 1,. is the contribution of troe sli tream n.-n the
s,-cyd'.l term is thee 2:ont ribti .in of the air :-i s3lace .3 Lv
-hF slirstream. nr dividing bvy "D and ujnlr!- the
relations = V(l + '.) and -V(i + 2a),

S/2 +

a (I i)2 1 (12)
(1 + 2)2

vL. E" is the in e ".'el -ne of -d;c ,',-h1 at tl]. pro-
,.ell i rla t (ref,-e:r-!lce ) d uc,1-e almost tv.i.ce this
xal. ;t -r-sll values of a J nc -.?L ti i, th r react -.rr
of t :-. t c s'-1- !. i ..y h -. str--.

t ":a s 3"-r :.ari.-" r thL. t t i.f: fftti v_ sfde ,i.d
In the ofl.Ler of th.:- .ro -11.rr t < nd i. d
is th'i .'.o'. ?nert aar'- lel to V, t' .t ,

dV' = ,. 'I ) sE 3 1 ,,8 )

h1- v1.'al .le of t'-'r. e.-4ui t ,r, (l rra,; ue *I ntrod ic-.d
an1,. thq l' t nr, E = + + ,), Ir( m r E-rr-l )'oin -i tu".
th:-'O ", -... t 1 t': te T, t. TI- re r Es c.ult :-

f':. 'i ( ()-' r,
L T 'n + a)2 1,a)

f( + a) + + + )( )
1 + (l + 2a)2

2(1) + 2a)
f (~) =- (15)
1 1 + (1 + 2a)"

PaEtio JV' / for jawed motion.- As Va = V(l + a)
for unyawed motion, the changes produced by yaw are

_a dV da d a
+ (16)
V i + a 1 + a

if dV/V, vhich is c s I 1 i-, is neglected as
being of the second order in '.

Tr order to eval .iste da, figure 2 is first con-
sidered. The c:loanEnt of the effective side wind in
the d~lretiLon r-.pposite to the bl-de rotation is
v = ir E' Fi n 9. This ccmoonent a.ts to increase
the rlai yve w' id at the olade, andr therefore the thrust,
in qu'rdrantz 1 anrd 2: it acts to decrease the relative
vi.nd-]., rand therefore the thrust, n quadrants 3 and L.
rlor-- xa-tI:. the Ihn.fee in thrurt due to the side wind
is d-irtriouuted sinus ideally in It is clear that
thinl increm:ental thrust distri bution 1by its antisymmretry
prodl.ces r r-itching moment.

"o:nentunm onsiderations require an increase in in-
flow in q.ladrIants I ;nd 2, where the thrust is increased,
snd de. ercase 'n inflow :n quadrants 3 and 3 here the
thrust i3 decreased. The var2stion should be sinusoidal
in 1, and the sssumr.tion thrit the variationn is directly
pr;-;.rt'oi-.l t- tih radius is sufficiently accurate for
co!mutintl' t-he effectt on the side force. Such a represen-
tatir.n is illustrated in figure c. The analytical ex-
pr,? s sb n i

dv = I9

= !lr sin 9 (17)

where is a constant to be dr'termined. Applying the
mom-ntum theoTry to e'.-al'.iate thie itching moment M in
terr..s of 1he inflo"; rrodifications nrodiuce-d by the pitching
momi-nt givess

R ,2
M = f" r d9 dr r sin 8 (2 dv)

; sibstitutlon of the relation for d*v,

P 2
S= 2k Va
foo o

rV sin29 dr

-T"-rcn int? -tr" :tion,

(1 + a)TR


Vc 2 P3D
but, (".: ,.c .'tions (16) and (17),

dVa da
Va 1 + a

kr sn + a)
v /(1 + a)

a c

a (1 + a)2 T


vhe r-


Suirr-t ion over blade ir-dcx -. 1.- The component-velocity
Sncrer-rnts .'Ju to ya'., ha:; te.n obtair.ed in the preceding two
sections as

-j-= ----- f(a) fl(a) -2 in 8e
V (1 + a) k
9 If



dVa I
Va (1 + a)2 T k(1

where the subscript I' hils been added to refer to con-
'iltions at the kth pronellcr blade. These values of
dVq/Va and dVTa/Va i.ny be substituted in equations r5)
to jield values of r5'dFc) and 5(dT5). The values of
6(dFc) and 6(dTc) thus found may be inserted in equa-
tions (6) to (11), wii'ch give the several forces and r,;o-
ments the propeller c-n.elva:vbly experience.

The summations ove'r i indicated in equations fr)
to (11) affect only tLhn f-:,tor3 inv/lvirn sin P and
cos 9... The several l ct'rs arr, upon evaluation,

sin k = sin ,, cs -

If Ba- 3,

k---. C-

If B = 2 or 1,

sin- = -l cos 29 )

but the averwgr- over e is :/2.

The nonvanishi'ng factor sin' 8k occurs only
in equation (7) for the :side fr)'cr. Y and in equation (10)
for the nitch-it. morint "'. "he otther hy:)thetical forces
and moments that !: rht bW .rnr1uced by yaw are-, accordingly,
all zero.


"TEn the rel-ti _-n
\- sin2 9,

is u.--:, eq:ations (7) and (10) become in nondimersional

Y f n-t(a) f () A + c T" B'x dx (19)

2V=f (a) f(a) C'+ 1 'x.' xdx (20)

For -':plr'ty :he following aJr;iitio'ial abbreviations
c re introd'Iuc-.i:

o' = '

1 A) dx

1 /(l)
a"* = ii / -. d

a' -- .-P')2 d dx

di 1L

vWhenp the siir,l hv- oeen c!ho"en t) make a', b', c',
and d oositl."e .us'ntiti s.

Solution for Y and M for single-rotating
propellers.- ''ith the precedlinj substitutions, equations
(19) arxd (20) b- ome

Y = f(a) fl(a) b? (22)

= -T- f( ) -r f (a) c' 1 r d >

These are simultaneous linear algebraic equations
in Y nrid ''. Th-e solution for zYc Is, after

f(a)a (' o'b'c' )
S1 + o'd'
c (a 1 + o' -i a_ 1_

+ 1 + o'd

which miy be written in the form

r r f(a) o'a'
--- + o 'a

f b
AV' =- __ ( 4: )
1 + o'd'

ITumerical evaluation shows that thie denominator of equa-
tion f23) does not differ greatly from unity; therefore,
Y is roughly orooortional to a'

Similarly:, the s-)lrtion for r, is

r f(a) a '
2 + o 'l f (a) (' + C'd') o' f a) b'c'

which 1.1?i' be put in the form

IT f a) (
L = Tr (25)
1 + ~ o'(' a')

w ere
m = (26)
2(1 + c'd')

'-e relat ve -t! t :tl-'si of the quantities are such that
Mv is ror-ihl.; orooortional to c'.

olutnon for Y 2! V' for '.lal-rotatin,-" oro-

ro t r.- o. -:~ 'ilers. With du-al-rotat ir.. :oc3llers the
asynmnetr; of the disk loi--lr:C, which for a sin-le-
rotatirg nr.oe1ller no' the 't!:,'n moment due to
Yaw, is o .-!itely ~-r'o i-:a over the ir-.nt and rear sec-
ti~,,I-. TIF result':nt c"er-all dls' losd'ing., therefore,
is almost .:, ,'r-trical anr, .ves rise to a neg-ligible
r, moment that Is,

M c= 0 (27)

The 'nrdiction effects asscIatpd with the respective
iss-3o~l~. r~, asyr.!- -trles of the frlnt rand rear propeller
rctiorns vcry near-ly c-n':l even thou, there is a finite
sen.,ration bEtt-'..n ti two sections. This faTt, vw'ich
nsy i- r -.3 rde,? as a con-q'.-lun'-e- of the relation (27),
is repr'e. nr-d b:v oi'ttin, = 0 in equation (22).
Thr result is

f~a) n a'
c ~ (2C)
1 + f (s) o'a'

This equation offerss fromr eq'.iution (2), which
aonlies to sin-le rot-tion, i:n that unit:y replaces the
Ir- r tr.lt7 ----- in tht- dF-enoinator. The
a' A'
side-fr e coi'ffclc'r.t v is therefore lIrger in the

case of dual rotati-rn.

.'itn data for conventional

nr:'pel -?ls, the incre;:i7e averages about 1 percent and
rerche .'- r:i.rcnt t.t low blade angles.

The- ine'rea;: in side force ic due to the lack in the
d\:al-rotetin-g r. pellr of the asymm.etric distribution
of inflow vloci-t'y across the dilk which, for the single-
rotating propeller, is irnduced oy the azsyrr.etric disk
lor.ding. 'l.e inflowv asynmtetry is so disoosed as to reduce
the c:han.e in an ..le of attack due to yaw on all blade
c-.1 :1-:! ts. 'sIe behav'lor is enalogous to that of downwash
in r--educing the effective angle of attack of a finite
wi ng.

The inllvw -syr -:":etr': is not the only effect analogous
to do.'nv\ash i '.ing L.eory: the side,-iash of the inflow is
anoth-r such ef'-rct j::nd -?srves to reduce the side force
,still furtnc r. Silv'-ab 's, however', co,.mnon to sinzle-
and *juLl-rotatin,. _roc-llelrs and affects both in the same
'sy. An ':.*:smilnLtion of the steps in the derivation shows
that the tern f i) o'a' in the denominator, the

absence of v.-..ich .ould increase the value of Yv is due
to the sidevash.

equations (25) to (2'.) give the stability1 derivatives
of single- and dual-rotating oropellers with respect to
yaw, but the results are not yet in final form. There
remain t'.iA evaluat.inn of ', U', c' and d' and the
introduction to' a f-ictor to ae:ouLnt for the effect of a
sinner and an-'ther fact-or to ccrrectL for the assumption
of unifor-:, loading )f the side force over the nrooeller
di s':.

Explicit reoresentation of a', L', c', and d'.-
Equ tions (21) show a', ', a, .d' to be integrals
involving the funct ons 4', 3', C', and D', respectively,
rhich areP i ifbii, d I, equations ( .) in con iuncti:,n with
e ,.qnt ion: (1) ard ( ). The quantities A.', 3', C', and
D' are, unon c vlu ti ?n,

A' = sin + cos (29)
a' =

3' = a cos c sin 0 + cac )

C' = ca cos 2 c( (sin $ 2 cso o)

D' = c o c cos
to oin / V

if terms in the coefficient of profile drag cd are
neglected as be'n"- sr.l in ccoa-rison with the terms
in c The electet of o is valid -n!.y for values
a o
of not too near 00 or '0.

'rom, fi-..r- 1, p = + a. Tri-n, for a in the
unstallcd rrr.-e,

sin o = sin 3 cos a + s'i a cos 0

Ssin + + a cos

el sin p c sin + + c a cos '
0 L L
a a a

= c sin + c cos 0

the rihht-hand memrrb-- of whi ch is just A' in equation r29)
This reltton provides tne irmpor.tant-result that, although
both I' and c dee.?r.n on th' inflow, the slipstream
rotation, and ti!e value of V/nD, the function A' is
I nd.-pend.l.t of these quantities snd ueperDn.u solely on the
geo"Etr-ical blade an-ile po. 'fhs relationship le'-s
dir-:ctly to the intr-rretation, to be established presently,
tnhrt the effective fin area of a.oropeller is essentially
the projected side area.

The i'.tro i ction of p does not succeed in sir. larly"
elinminating / an1 c from B', C', and D' but does
result in a sirplification in 1' and C'. The summnrizer
results arc, to the first order in a, with D' left
un hanged:

= -c, cos C C5 csc

C' = ."s 3 + 2iC csc
( C 0 '0
a sin 0 I

The Integr.-ls (l1), in which A', B', C', and D'
oc,? .r, nust ni-: te ev'I~iua .t:t o-.n Fubsti'tution,
S 1

a 1
E' = / sin F' '

b' = / cos Idx / j dx

0 x0

7 1
S- / .

C 1
d' .

t_2oj .~i
f-in 'I

3d x + Il X c -Sl A dY
X '

-7 -
S" I
dx a- :x'a cos dz

wh e r c


and b is the -l.a-le widt.l..

.vsluti i *n .f Na' and 0'.- Thc integral a'
is alr-er..iy its sr3 nle' t forim in eqisLtions f(30), as is
thi: fir-:t inteF ral *.f b', which is identical with the


first inte-ro7.l of c'. If the first of b'
is defin r-.d as b,,

b' = b b I
= b! + 2

Wvh- re

b1 4P1.:. Ja -


b lu.: co r/ dx

In the attempt to evaluate b it was foun3 that,
if the blade section corff'cient of n-rofile ;Irsr and the
rotation of the 7'pstream are r-.-ected, the thrust
coefficient is

S (1 + a) ro'
I .T L


If an fa"er c-e viae of 1 + a over the d. sk is us-d,
1 + a rn bt taken frrm LndJ1 thi inteAral s i n, Pnd

J Ible
e G' 7 ( 1 + P)

But, by the sirm'ole mominTtuw theory,

7 T, = a'l + a

There fore,
T 2a
2 o' n

A grjph of the variation of 2a/T- with T is -IivJn in
figure b.

An 'roximate evl uat' orn of d'.- The contribution of

-7 1 c /x1
.' = c sin dx X ax acos 0 dx
T7 0 ., sin 14
"0 "0

to Y is small. It is found, by usin-- the largest
value which a nma have "'.out causing stalling of
tnhr- blsdes (about l1/ rad.ian), that the second integral
can oe n:egle?-teri, with the result that

C 1
C T a 1 s y
d' x/ r

IIote thlt C involves the inflow velocity y and the slip-
stiram rotati nial ".'loit.y. These velocities, if assumed
to be ronst anit over the prooel er di'sl, na; easily be
related to T. and ,, reso-c tively, from morent'rUm
consider ati os.

Cir-'es of J' have br.o:n omputdr fcr a typical olan form
( ?Har. lttn- tand rd oroocel lrr 1 55-o) and sr-e .rrese;ited
in fi.2ur; 7. This cart make's use of an altered notation
intrcdu.ced l'atr in tr:.h report; tn: ordlnate is the
quant it

17 =7
=-- d'

and Pa ctarmeter is the solidity at 0.75R,

: O.75P
D 7 5 R^

The abscissa is V/nD. Tihe error in computed side force
due to usin; this chart for olan forms other than the
Hamilton-Stt ndard 3155-6 should be negligible. The
chart is not sufficiently ac.turate, however, for precise
computation of the pitching moment du,- to yaw.

Correction for noniu:-form distribution of side force.-
7Th i ndiced sidewvash angle E' a. call=ulated ir tlhE
foreolnr, is based on the ass.unotions that the thrust
and the side force are each uniformlyy distributed over
the propeller disk. The error in effective averqs -
sidewash due to the as'on of uniform thrust dis-
tribution can be shrwn to be small; the error due to
the assumption of uniform side-force distribution is
ov:rec'-able. The effect of this error on the tvrnluted
side force is small, but not ne-li*lble.

The side force is actually distributed over the
*ropnller disk nearly as the :ro-iuct of the integrand
of the most important term in the side-force expression a'
and sin2 8. The integr-sr,d of a' is proportional to the
blade width times the sine of the blade :ni-.le, which
tends to be r-.eatest t, -ward the bl-,de roots ,u,-- to the
tvwst, and iin2 0 has m-ix n.ui-at 6 =: -')0 and 2700.
The side force is Therefore concentrated near the blade
roots !':.d tl;:n the Z-axis. Tn calculatin, an effective
av. ra,I. of *.,e part of e' due to the side force, this
distribution of the side force is taken into account by
using in perfect th,- int'g3rarnd of a', v:hlch is L sin p,
times sin 8 as a weight factor. The Idetailed treat-
rnment. is : ven n Ao.jenlr: y A. There is ob'taUnd for the
effective average of tlh ind.ucrd sidewashs angle
2 (T, j+ kW11)
= -(5)
(1 + a) i1 +
L (1 + + )


1 r dx

3o n
X x.0 : n3 1Y

is a correction factor derived in aooendix A. If C'
is inserted for e' in the analysis for side force and
pitching moment, the corrected forms of equations (23)
and '2) are, respectively,

Y T fa)o'a '
Y --
S + kO'a'
a' 'T

n f(a) m
S 8 1 + 'ao' (a' a')

where the abbreviation

f (a) kI
k a = 3 s 1

5 ,L sin )d


has b-en used. The factor ka ray be called the side-
wash factor.

Correction for augmentive effect of spinner.- If the
spinner-nacelle or soinner-fuselage combination has a
fairly : large fineness rati., the circumferential component
of the side wind is speeded uo in passing around the blade
shank:I (fig. 8) by ariproximantely the factor

1 + K


sinner radius

.' constant (1 for fineness ratio o; 0.90 for fine-
ness ratio 6)

This local Incr sce in side :*.inr is equivalent to an in-
crea&- in the .4:.,!3 of' ,;.'w in the same ratio. 'iTus
at radius y' the effective angle is

+ 2 ~|
9,(x) = i) + I

The effective avezsi yaw over ;-e l dik is octte'!':-:
fr-.1 the conrside-ration t.-t dYe is nearly pr -: orttonal
to the int-- r-i: .-' i sin p of the dominant term a?.
.-.' :." -tely, therefore,

Y = /I 1 + 1j. sin (6(dx

= '- ,e / sr" po

',ihere r is a constant. ..r rnc, r ,

S j a sin l, dx
1 + ---- --- (56)

/ sin Iodx

1.c .r in t_ to this r;.-:l t, if t -e or.*'?? r is
rqlu n: E th a s'.lr~ r i l -: ;'i -' ,: i '-. re ssoons
for rid rr. r e and it 2.i.;. :r-.,--r t .:!-, P:.1 L.-- :r tiplic d
b:,' t.e --tai it 1, an :.. r b t,-.rr i the r1-in r
f .:. r. 'he valic- sf is o tLE -)L.r of 1.1; and
var,.eF 'li, htly v" iti f Id' ne-l.e.

Se' de f : ti n.-.- Tt s '.'.-irth ..bile *o in t -duI"e
certain nr:r j- t 'r.s at this -olnt to lit the i -:l1
equation!i in boetter- form. Tic ori.inal ide-fi. -ti t Di vwere

chosrn szolel with a view toward clarity in presenting
thje "eci vti on. Trnc pric-cip'3l change 12 the replacement

o' = B 1

which is orooortional t3 the solid t.'i at 0.757', by the
actual soli-it:, 9t 0.753

0 = -- 0'
5 rr

( 57)

This change entails r;llacin all. t:e- integrals occurring
in ti F equstitn by 7T/'4 times the former values,. Tus
a' ic replsac-d by IT, bl by d y T nd L' by A,
'"ie re

T = r

C 't

a -- A'

In adl'i t hin, f olo'r.,ing definir tions are intro-


y '-

; r.d

0 T T ---

iT C

where thre propeller disk area

~ 2
S' =

The s,::'. s 0,,' a.'L a have been so c.-.osen in rela-
tion to the conventional Side-force and pitchlng-moment
coefj.'cients of an -'lr,-lae and QCm that con- is obtained thr.ui,.h the relations


-, p
-S y

C '

wh-:r i th- v in- a ~es and is th1l '.. g refer-
er.3- c .. r '!ot- t ,Lt in 1il '. L. for0 nr iss
me as f,'_ ? r. 8 ,' T ;..n s.

,r --.-'- T' of -'D f ,-e fo] .?. -r C :' "lity.- I t
s .3'" .i erdi" t'-..' '. :r, .,.c'l e ..:cr-r'. ti.i n for
c r: r: L'L. -it" is o~ tai.i:'e, L- S' l 0i, ':-.l: F ,i e lorc,
by 1 "'-ere i. r .:,td to t L- .tr -ni "'ach
ry ix i -i" T ..
corr..ti n. is v'i.l J )nl7y be ,,,; T... :rlti:t l ""ch number
for f:.;_ :.r"E'j ller.

S irr-arized. effects of -'aw.- With the new definitions,
the C.d--force dr-i'vativ. for a sintlc.-rotating rro-.eller

1 qS'

k ,rfa) CT)

+ k ,fl

and the side-force derivative for a dual-rotetilngr ro-
peller is

C' q3
qr qS'

1 + kl CT,
+ C

derivretive i s

crr" qD-

k, f(I) nm
1 + "-c (: -A

and f'r a r-g propeller the oitchine-rromrent
der v ti'-' ;i n-e.g!l bl].y small

The e der-ivtive may bF correct' for corn.-
cre3ssbilit7y jy divdir:ng y '. T]..e sane cor-
c e
i-ecc. on b-t',- be anloe to the p1 t2'.iing-oro-.ent derivative
but 'lIth less accuracy .

'7e i&uintlties invol-:d' are:

STinner factor

k =1 +

i / 7 L Si 0 dx


Si:>.-'r..s factor

/ .,- sirn2 p^, dx

^ Ia) : -
a *> / -
c| sin .. "
'\ -5

whE re

2 (a) =

2+1 + 2a)

1 + (1 + 2e)'-

Inflow f ctor

+ c----


S(I + a) [(1 + rt) + ;1 + 2a)=J
S( ? --+ -- 2-------
+ + 1. + 2)'-
L 4-

r( '

( 6)



(37); c t '.~,



1 I.
T C x


-z 7 C7 q*ip
T_ c f 2 dx (US)
2 Sl 3sin ,

(: J-- 0) + 2T$)

c(l + olT)

cI1. + 1
eilm += (-)
21 + oi )

and. i.-i cqulLtitoni (c.) for a i; elle finenress r&tio of 6,
r b,
z' ld, for a fines ratio of I' = 1.00.

The en~'rts of' fi3 'ues r,, 7, 10, an d 11 ure provided
for d Iteirti L.n Ig "/'T, T f-I s it), respectively.

"e'uired a:3'airsay of 'j ka, and A.- To the degree
in whi.h .c3;:r: ri. sn vitl-. ex'-stin *-::er'i ments Est'Iolishes
the 1 t-.- ur.-!.s of the st.d -force for,:.ulas sbout i10 per-
cFn't 3"..razcF err-or it is su fi z T ernti-' pcclur-te to use
the n..-'-: values -.!. for 1ka and,for the usual-si ze roinner
xs = ,. l ), 1 .1, for .. 'T-, the sS',e asc uracy, the
tern-s '.1 J r ay of minit t fromrr n d 17 mIay be set
equal to th'e ever.eE 'v':..e j, v t-!; t'.2 result that


1 + ".o

Avsil1bilit of chkrrtE of ~'id-force rPriv'tive.-
nrn reference 13 is presented an extensive series of charts
C,-, ,ut:e :1 frc.n '...ationrs () to (4Wl) for two conventional
propellers. Te. derivative 0 is given as a function
of Y/nD, for bl-de .' isra .-.i,' from 15 to K'. &r,
for solidities fr:wm two blades to six blades, :-th sin i.e
rotation and dual rotation. Tn reference i!. is presented
a method of extr-,iatlon tel- ;. this set of charts may
be used for deter inr'.a. "' for all conventional oro-

pellers without resort to the o~" 'nal c .-.tions ( t>) to
(44) .

Pi tching-inorent derr vat ive. nIre rY csl evaluation
of 3.-. t.:l~ 77 r1T t -. '-' ":o rent of a ts .- e-rotati r.
pro':,eller in is found to of the or-:r of the moment
produced 'a ore ecual to the la *' force actr at the
end of a lever arm ":...- to the :3;o'ller i
moment is small and has heretofore been :-:- .1ecti"' in 'ir-
craft stabolit, st!fcE. : t that *'e effect is a cross-
cou.lin between .."; '. ;: pitca.

T?.I dusl-rotatf.. .:-" .ler de-v lops no pi tf' 1Ii

Fr. 'iler Subject to Anulc. r "clocity of Pitc-.

T.-tlo /Vo for uLar velocit, of pitch.- Th
.nL. l-: r vel C ty : -tcn : r:-> i.: t contrilbit-' n to
the rotational vploc,:; in the plane of the .-roe~eller
dis' It is krorin _"-":, Clauert's ,,r-': (reference 5),
Iov'v-.r, that oitc.l.. 7 es rise to a sa.e force to
, oitr n ,,..n.r s ? 'S- f r,'- ",,'* _-'' .sh
that ff .-t r ',, i-: in t!', 3 .- o LI ': / I" i- er.
The an': .. : : r' -. :- S-.. s t'-e L .. -
' -r t ,f_ t ? -,. ,: fr .-., --, -,' < ,

'' )- TT

+ ,. .. '
1r ,) .-.- (15)
1 + 1 + I-

Patio dV fr engulcr velocity of pitch.- The
lirEct r:czr'-mnent, cus to iitchin in the axial velocity
iT is qr sin 9.
The induced increment Cdue to tte rifore-mentioned
pitching nomrent is, boy quation (15),

(1 + n)

The total incre-:nmrt e".
*:he irnuced inr"i mcnts.

d'v Fin

rc ss a-Ls f ro
r.::pre1ssons for "

is tb. s um of the direct and
Th prefore

a 2) rr :
21.' r _


and V T,. U'ron introduce np the

eque t r.s (E6) and '!7), the equations that result here
in plaza~ of equations(1 .) and IdL) for the propeller in
'a&V* 3rc:;

2o LV.

e *[qDx 1c -ex
( r, -- A'+ ( 1a) -+

9 1 t. 1 '' R Dx l+l x
2%1, =B -(O,, j L -''l a) ct -- +-- "' ,xK

S- lution for Y, end ,. By using the abbrevia-
to.,ns -i equationss (21), equ)ati-ns (U1) become

)r c FD 1.
Yc -f (a) a + a) D
C b& 'V

2-A -f (a) C( + a) q + 1 a

-hich are simultaneous linear 'ebrL.ic equations in

and ". -e solution for VY and c"
s 1 I 1, 1 fi' c ati on,

= 1 + a)
i,' 7



is, after

2 -_

+ f a)
+- o(11

- A) (l + GI)

- (1 -_
_c qD + T +
1 +

Side-force derivative C..'

i.. -It ln,

deri vr-t ve C, 1 .- ..-force -. oitch;-i,--moment r--

ri,.t-ves r",':, h' defined as follows:

CJ T ..
q 2

= ( 1 + :: )

1, (r_
4. CT(1


(1 + C! 3)


m p
q V2DS'

/< fl(a)
1 +
*I a i of- o OS

2 flia)
1 + G(I A)
0 1

0ou1h sa~'roxi.>-ti _o)n mrray be obt:.n:-d by omitting the
induction ter.ims that is, t:e terms CLue to '.dewash and
to inflow asymmetry. There result

C + a' OTT (49)

C = U: + a) --
Cq 2

Crm.neriso. of -._ ..le of .,'r, with Langular velo.s:ty of
oi tch to s.-m s'-rd force.- To the same roauh
a ..nro,.mtrn t n r s *, aL *,. f T ,

,a' f a 'a s '

Ti: rst'To f ; to qD3/2V t OroJ'lce the same side
force : c!r'- -ore


(1 4. + T.

f(a) k T1

'.- T

'his ratio is of the order of unity.

.". Y. ': ;. ..1T KT> f< .i -? ,i- tJ -) itchln; .- The
raxiMAr .-... .. f 1 ,j t V' r.- oc urs,
for a given bl':. --- e se'ti:, when qD'-" is a
::i.: "'sY" -. q V.."_!'' in install d' flight s de-
terl-,n.d by the rmaxrimum normal acceleration that the air-
plane can rvel, w" cch is deter -: : ty the mn.-xi:r:- lift
coefficient. Tht normal .acceleration is

a = qV

fi ', *..i ch

qD = (51)

.t i' s- r. tl-.e max1i:.un. norm'.l accelerate on almax
cou'ln. bp -.t' at the toor .:) sn ;nsi lo') The
relation -: .

t- V' +- "

rrA, -Z 4

The v, .1iue f a i/2 is gre atc.-t vwn V

Tf the disc'isE.T !r. i.nite31 ior the present
minim-u, ,r- e,,nd for lev-1l fl-i>ht "V

-2'1/S Vstall1

Prom cquat-ion (52),

nmlax 2
vl t V t

*nd th r;eforf f'romr e -is o ti on (fl),

is l] ast.

to the

/CiD X


A nractjcal u I-i:Ce- limit to (qD/2,'V) at the
stallin c spc -:d "~'..1.].rd f aff.,rded by a hypothetical fighter ai-
plar. haviii the fOllovnrig charac.terist ics:

stall = 5 mnn
= 113 fps

L = 12 ft

The ',


=2.2 x 12

= 0.352


By ;1 ',t- on ( h.) the a..-le of yaw', in rrians, that 'o1.1d
provide the same side force is ',. .roximately

1' max

Tf a r":.r.,um blde ...-_le of 15 at stall .I :'-.: d is
ass-'r .., the ratio I ._~/ 1 i s .15 for the r.-"r'-sentative S :.-..:'j,. .3H er 5155-6-. Therefore, ( -V)
would be I i".valent in p:" '' s -e force to an -:,.-le
of yaw

t -= -1.13 x ",.);

-0.'6 radsn or -2.1

The results side force would oue : .ite sall.

'nv, tlries the -:,-' --:" .. value of ( i:,'.' ) is
obtainable i. .rr'., the .:, v i-hc. inv ... ':.. stal'.
If the s2 n is excl.ui2ed ifr,.. consi- ration, th:refore,
the ,- :- ral conclusion to -.: the .r. le
is T'i-:t even inanan eYtreme r ..-x~ver '" sde force .;. -'.
to rite of niti:.i. is e salL and in all r-
rm.neu": rs this -ide force jis 1-b e.

d L' 1 n1 "'r".... .. t. "t : b l

erf t 11 r S e111r t. -a o ri r*: r.~. 4 '
S : ', T : t -. i

f r r.-l 's t. .~ -- t .- n.-:. .. t .
i -" l 1. ". .... i ", '7 1' -: ,--, r '"._. .. -: :. ; ti ,- = :- ':
of t; s n, *. d i r. ] or r. r .r ..:'.. -- 3 s :.-: i 1:.

.ore- -i IT r j. :r ',- -.- t or ** ,- *. ,.' pr
velo1..ties 1 ': ..**' .,.; ,r-.-r Lt v.^s )i' t:xh- '.,r.;-,- ,?,ld r s.
ar.guila. tl .:.ti s :'F ", t -. '-.I, f ~-.' .- or, a :..rIr.- l 2er
due to ys.- or.: .11-:- thare du" to it 1 :., n-:-. i-: ble
exceot n t i 2,-,ir'.


Cc nce t of Iprojected side arFa.- ThE are-. orojectrO
by a i;:'rpeller .bl. de on a nlar.e thr-ou-h th-e .xiz of ro-
tattion and the axis of the blade is

/ b sin o dr
.' r

Th, a svc-r.-e area nloje ,.ted by all the blades If a rotating
nropell-.i r on any l'ne through tohe. t- axis of rotation is
the projectedd s2de area

= / sin dr

whear b in the nurmb r of ol.ades. Fr,.m third s rri~tTrn,
it con b ,:ctablts.ed that t'.-e orodut o r. ay be ex-

rhere is tn rr-o-lle disk ir-s. Tni s I

which -fih i.r-.. s. orn."r nr n t I.-t;, in. the expressions for the
sid- ir.orc- '-. i va ive s' is propor-tiona to the

oroje-tcd si.e ar"a of the rroeller. In rieerence 17,
I! i ter-nei. the "side-area index."

rfr-?t.\v: fin area and aspect ratio.- 'nasmu~ch as
D /S. is the aspa3t ratio A of the projected side
area ,,, it is also true thet

i- (54)
a '= A

Substitution of equation (753 in the num-erator and equa-
tion (4) in the denominator of equation (59) gives for
a dual-rot tlnz nvroe-eller

6 vI tj
f(U) q ..

i(a) I-

r(a) q Sp

1 +



. ka =, 0.l. on the aver 'e -A'LV


!or comparison, the corre'-.' .l';-nt e': ression for an
actual fini of the same aa ea ijrl c.set ratio, at which
the local d' n'.: :c pressure is f(a)q, i.

f(a) q +


v'h-n the lifti!r-ljne form of riect-r?.'o correction is
u-ed. "-i r't'n,; k, whioh merely accounts for the
favorable intwrierence oelwe,:- i r oincr and proellUer,
e.1.vation (55) can nL written in the form of .-,ustL'il (56)
by intr.:'iu ?in" s- effotve -i-'ect ratio


It fol "':l's th-t sa cual-'-tat In#; nrora eller ini ysw
acts like a ;in fi w area of the I o).:-! ler, tL'e 'f-fec tiv.e aspect ratio is

seoro-irrrat. 1,y tt'o-tni rds the sIde-area aspect ratio, and
the locil Cy,'r.--m~i pjr.,s ur-,. is f'a) ti'lz-s the free-
streani vj u1 A snigle-rotat n oropellr may abe sho w
to act sim il arl2 o,'t th-. -.ff.:ctiv'e "scct ratio is m:rk-
edly l:--s and Is not so s3'.ply -:yoresFed. P. mean :ifec-
tive nsoect rat.o for both sinTg- aid dual-ratatin-
propellers is about 0.

Fff'eltive d-yn.ra4.c p-.P:sure. .' the definition of
a, the e::res:io n V'Il a) i2 the axial wind velocity
at the n-roreller dis'-. Ac r-r.,ll, fl + a)V is the
dynamic ,rc : srer at the ni-'ro ller dis-:. The rre ss:.r-e
(1 + s) is o'1ly slightl- g-reater than f(a)q, til
effective n nnic r ssur *-f -q'..t',r. (o) Thus the
equiv-le.l t fii i sc ribelF in the. rre Piin.3 paragraoh ;iqy
with sa,.l c-rrr- t'- rr-garded as sit inted 'n thn infi.o'
at the .oo-l ler di, '-: i si ucject to the *'orresoondl nJ
u.m:- nt-:d .-,' nc i. c o. u-"r 'u,- .

rr.-i. soi'L ,If sicle forCe -)f sin, e- and dual.-rotating
pro'ell. er .- It has Lren r::;rint.eA1 or n trhe -i. ucvssion
C oroi:';::T-'.i th:, der vatmnr. of 'Y i? forl dual-
r' t tn, .ro.-i e rz .r. a v: t.h t the du91-rotat&-:. pro-
Cellr a"-.'rse, l. ne-2rcnt more: Aide fcre_ than the s 'ngle-
rotatin ,',r-:, n 1i r "nr. that the in:r-se "escncs 52 per-
ce-nt at L-a bladjir anles. The aetrail'd e:xr la ation is
-iven i ,he s.e. discIs :in. In b: ief, dual rotation
eliiiin tc E certain i :-A.-ct r.Cif efl cts C es3sc), te r witn single-
rotation- o the .i eal-r.ot -.t ,. ::r"opell' r acts as if it has
a c.onsrid:rably 2iciher &3 soct rato und' therefore develops
nore side lorce for the 1 sre 7 'iditt'.

S"sxn tu.Je of oi chin morn -- it.- It has been shown
that ;, g, 15ve's rise to zero pitching r-oment for a du.-l-
rotat-in rron*1iler and. to a. fl. ite pi thing mrr-ment, given
by 'quiatio, t',ho), for a Kinele-rotal4n:g propeller. The
nuTmer- csl ev- lu't ionI of cq'.:ation ('I') f r typical caSES
sho!,is that the -itc1i nrw m:nrent is of the order of the o r.'Ir:urd by a force- equil to the slide force, acting
ct thF end of a lever arm equal to thr. propeller radius.
This cro'--coaunl Sn betwe,:-n ?itch andl yaw is small, but
possibly '.t i C 1 igi ble.



The results for propellers in yaw may be applied
to Oropellers in pitch from co nsiderations of symmetry.
The normal-for-e derivative of a propeller with respect
to pitch is equal to the side-force derivative of the
sarre propeller with respect to yaw, and the yawing-
moment derivative of a prooeller with respect to pitch
is ejuel to the negative of the pitching-moment derivative
of the same proorller with resocct to yaw. These re-
lations arr invalid when the propeller is in the upwash
or downrvash of a winr. (Spe reference 15, p. 12.)


Exoeriments of Bramwell, Relf, and Bryant.- The
exoerinients of 3ramwell, 7elf, and Bryant In l14 with a four-
blade rr.-del pronel].er in ~aw (reference 15) are worth
notiuln because the e:-:ner..rntsl arrangement was designed
specifically for the problem. The balance was arranged
to yaw'v with the propeller and to measure the side force
directly with r'-spect to nody axes. There readings were
inherently snimll in comparison with the forces being
measured. Tunnel so.-ed was calibrated by comparison of
thrust curves for the same oro':,eller in the wind tunnel
and on a whirling arm.

A calculated curve of C'/, which is the same as
Cy,' for small values of ,, is compared in figure 12
with the experimental values of reference 15. There is
included for further comofarison the theoretical curve
calculated by risztal (reference q). The curvr calcu-
lated from the formula of t'.e pr.-sent report aoenars to
give some"-hrt better agreement than that of Y.isztal but
the improvement is not conclusive. The or!ncipal objec-
tion to "'isztal's formula remains the labor of its appli-
cation rather than its defect in accuracy.

Lxoper.ments of Lesley, ,.orley, and loy.- In the ex-
periments of L:sley, 7.orl-:y, and "oy reported in 1357
(referEnc t16), the nac',ll. was shielded from the
air stream, with the result that only forces on the
propeller blades were communicated to the balances. A

.-foot, two-blade rroocller used. P'easurements were
:rade of si- components of the air forces on the propeller.

Calculated c-urves of C '/. are compared v'ith the
experimental valoies of reference 16 for ^ = 10 in
figure 15. HIot that the original data of reference 16
were presented therein vlth r-es'ect to wind xes.s, and the
data have been converted to the body uxes of this report
in the presentation of figure 15.

Experiments of .unc.kel.- The most complete experi-
mentEs cn yawed2 prooellers the only published experi-
nernts on full-scale oropeller.3 aire those of Punckel (ref-
erence 17). Riunckl tested sLn:le-rotsattin propellers
of t--o, three, four, and six blades an.d six-blade ig oropeller. T'he di amseter wzas 10 feet. An
attermpt V.,as r 'de to correct for the wind forces on the
rather 1arge unshielded nacell by' subtracting the forces
and moments measured with zero ya,'.' from the corresponding
forces and morrents measured v',ith yav 3t the same value
of V/nD.

Calculated curves of ,,'/,, including a spinner

correction, are comp";red in figure 1h with the faired
eyperimental ,urves frcm reference '1; for 103 yaw. In
reference 17, rs in r-ference 1:., tnh c.ririnal data were
nressnted with res-ect t.o '"ird a:xes and the curves have
been converted to th body axes of this r.-nort in the
pr-sernttion of fi:ure 1i. In figure 1I. the unpublished
xoe rime tal&1 points for the single-rotsting six-olade
propl]-er are Dresented for comparison with the fired
published curves as converted to body axes.

Accuracy.- From these several comparisons of the
theory vti'th rxoeriment it appear that the average dis-
vreemnent is slightly less than t13 percent. This
accuracy -;s of the order of that obtainable by the vortex
theory for the uninclined niopeller when the number of

The pi op'ellers of reference 17 w-ere actually tested in
pitch rather than in yaw but, inasmuch as pitch becomes
.Ei'/r uron a '?0 rotation of the axes, this conv-rsion was
mr.&? to keep th` discussion con-istent. In this con-
nection, a vertical force due to nitch has herein been
called, a side force due to -a,.

blades is tacitly assumed to be infinite by the omission
of tne Goldstein correction for finite number of blades.
The same assumption is nl&dLe in the present analysis.


The foregoing analysis of propellers in yaw and
propellers subjected to an angulir velocity of pitch
permits the following conclusions,

1. A propeller in yaw acts like a fin of which the
area is the projected side area of the nim eller, the
effective aspect ratio is of the order if 8, and the
effective ivnamic pressure is roughly that at the pro-
peller disk as algmented br the inflow. The variation
of the inflow vel.'cit:y, for a fixed-pitch propeller,
accounts for most of th': variation of side force with
advance-d! aieter ratio.

2. A d ,al-rotting prop,.ller develops u o to one-
third more side force than a 3: il:e-roatting pron.ller.

5. A yavwed single-ro.tating propeller experiences a
pitching moment as well as a side force. The pitchiin,
mno'.ent is of the order of the r.o-'ent produced by a force
equal to the side force, acting at the end of a lever
arm equal to the prooeler radius. This cross-coupling
between pitch and raw is small, but possibly not neligible.

4. Pr-n'c. 1ler forces .uie to an angular velocity of
pitc' -,:r 'aw are ntegligiblv small for the angular ve-
locities that r-.av be re:'illzed in maneuvers, with the ex-
ception of the snin.

Langley IMemorial Aeronautical Laboratory,
NatiL.-nal Advisory Conmri'ttee i'r Aeronautics,
Langley -Peld, Va.




If the;ition that the side forc-: E ,inlfcr'rin:7y
distributed over the oropr.ller ,liski I" j sband,-r.ed, it is
n;c'.ssary to proceed differently beyzndc eq. tiins (5) in
deriving Yc. ,or the purpose of obta' niL.g on ffl.tive
avera:&e iniduc d s.icdewash, it is ,:r-rr;is-'ble to rnirl.:.*t

thI small term B' in :qu.-tion (.) which eives

S(1 + a)2 b -II
4 70.75R va

An equivalent differeniti l relati'-,n ic- t!-e time averare
side force, divided by F-D2 on .n eleerent of di1,k
area x .. dx m',- be sl.lstituted f'. r t'.,e s'.ir...ti=jn of
equation (7), as

.d dO = B5(dc) sine d -- A-)

The fr.ction- -?- has br-en ....-'._ n to or_ r- iv.,n '-,'

(4 c') sin 6 (1)

wvi re e' is the local induce.: ai,.le -. side'.s. h 't tI-L
or'-.,eller disk. ,,r:.o ,n e:*u; t_,n (A-1), '..-2), an-: ly),
usn AL' = sir 3and: that (i + a) is
r I
consta.-it ,'-' r ti,! r -'\-'---3

d'," c, (1 + : s'r' ~r sin ? / Cx
d9 $-


from -ti ch

So'(! + a)2 1
y -./ p, sin Po dx

1 xo
1C; s Ii n2 t s 1. n
-- .. in2 ai dxd

An effective ave; of E' is ob't.:' i by icf: rn.

S'(l + a)2
Yc = -- *') s.n d d(A-L)

from T1.:1ch the'i:-:-2 EI. sL. L S

o x

te l cal vl t :'is2 e l:n .r! t -. .. the
= --( A,- !)

t sinE s3 :..

average value usar in tL-e main t-.'t; '. c -o :,d -,f
one pe:a t Ju: to lnc s tde f rce e' tn. : .:. 'art ... to

the cross s-,Lnd 2o0.)on,.nit of .f t .ru c' + -"-" Ef-

f ctiv" over'ages *re : ;. i"gna t er.. 7",, .-..*..., F T:',- n

e iati-ns of th fcrm r f e '-qiu ti3n: (.-l; 3.1 bet-':n
-E' a-id e' and '-et'een ?' ani r'
7 i T"

The evaluation of 7',, follows: The product

Va ',, sin 6 hereinafter called v 6. is a velocity

romir:on. nt parallel to V@ but not ir cF'ssarily in ti.'e

same ,sense. In order to evslilat v. for use in

equation ( A-5) it is use- 'l t .. '"efni n trLt t;L:,- f

such that -- (. i- i thi t'me averarc of the
x 'Ax ,.9
incr; rent due t.? ":.-' f tr-,- '" i' m 'l'r-i r l for.? rin 1r: e : ni. t
of di sk 9 r': a dz d. "a'i:n! t!i Fi~ m l i g nn 3 -
ti on Lhat thi,- ..ri :h r-.: :, .. .e.n-..t .f tie ,:.rc-
Seller di?'-: sffE-ct s -on".,, the :.r flj'- r. t[:rr ;.'ri tr..t
el.,m n.rt ant d ~i-q-tirs I thi :-: rz;l r,.ral 1 orce t- iL!-. ratio, of
chan,:- a of --: ri .*c r r,.-:rr-n un v -;1. his rcE rr..- ..p c
far Ihi nd th-e rT-.n.Ier ,li ds t-- t.- r I : ?

7 d d = p r dr A.i 2v.. + pV r -'v
x )t. 9,

(se-- erit vati on of .il ti .n (1.',

2 ,
9.. 2 --

-7 Q, 1 fl

I- x -
SL +

S- .- A- ,.)
ii + ) + -------
1' I + cr_)

'vh' re

f -
c V-D7

V = v'( + a)

Vs =V( + 2a)

A,: alternative form of -: uatinr (A--2) is

----- sn 9 z dx d9
c x -x 60


S in9
x bx 69 x ;: 69

In equat on (A-5) the f-ct tha:t n', ,*-ch ;-- .nis on
6, is small cr"' rec1 ..'t aliilows tie royx. ae

k sin' 9
There k is a constant. Integration establtishes t .:
value of k as 7'c/r; therefore

c,' r n

3y equatic.n ( A- ) ,

x r /iy __r, d.

The re f ,


n E-

T .--- -.
Vt. A 7 i-

eo"uti n f A-5; s '-*i..= to E' in nls"e or F!

Thj jil v e -

V -. ( )1

(1 + :-) l tl

Terefors -brti a ti A- ) ? rL.: to C'i Th oves o
-, ,- n. ;.
______ ,_L :L Z L

S 1 + ) + /n d


'h,. int-: riion with respect to 9 results in

E v =
(1 + a)2 +
(1 + 2a)


( 1

)j.2 s
iJ. sin Po r1:

T part of e' '.. to Y, in uation (12), '.. ch
is bas.-. on the ass'uwt ion of .',:.orm c 'strilution of
tlr- 'st '.r:-1 vide force ox er the ropelle r disk, differs
i~or the exoresston for -', ven ':,' .'. ::tL on, (A- )
only in the absence of the factor


-: -
l-i- ---- dr

.o /


which i oq-ation (~l .). An mn 'ypis for Z'
to th-l1 for I'v results tn s value that dcs
3,r cLabl ciff r from trns .-rt of e, to
Squati on (12); th t Is,

- '.. lar
T in


I; I
" ".) +

I' 1 .. ) +

A_, ce ,-'d nel :, t'-.. -ffe -: ".', :,-.'--r ".- "-. "*-. ., "

AI '1 + 1 r_


whi :h is equet on (). f c' is r.serted for e'
in eiu.sti-tt, (12), the f3 tor a, ( (a u in r. io..--t: ns IL )
-, fa)
and ( 5) is r.laced .y k This is ta'- q, u..ntity .Les .iMer, 1 led the z-dewash fc tor '.'.rth the
valuE of r 1 Inserted,
1 -.,)2

Fv r.
'= ', (* '
r _, (35)
-/ 2




.''h. side-force derivative

pr:oport;onal to the Irn te3:rral

X1 t~i2-

is very nearly

[ sin o '?x


To a first appro:.imation the effect of c !-r-essibility is

accounted for by rerlac'ti b1 c/ vIere

V is the resultant se~,-i of '.re blide section at x

divided -;, the s ..
the subscript c' is
for rc ',crezsibili ty

1 of totJ,~, in tbe t''r.- strermc. If
-".,*d to is', entitieses correct : erl,


4A m -fecte h

A mean effective ach number T

( -1)

d fr..*: : the ri nation

wo.l1. a lso p') rc -.-. t cti- t : '. -

-~1 / -J

Equat.ion ( onr t'i t'tt._ t'l3 '- .' .: r"''l : t rL- of the
side.-for.e derivstv.: f r :- r .re: L.' 1 ti :, ct?.



I- r et r.i ini

accur-.te to ;: it

.- --:

f -


+ ----


I +
,- ",! ;

t I r1 i. o u, h i :*: r 7 17 x ^ '.1 t c n
l final : qu-i i:.r-. i 1-j ).

F' "

. in d-


By r :- :n," = t. r1.-ur'-s 1 t:s.,
ar: e," :* i: i ,

2, : f tr. -.- i : 1'. -] rotation"

( L- )


(; -L


-. : i r

.3- ,.)

'fhF ,: ter n '.iit -,-.In ,f C, rcc, ^ s q iccllo s:

h. equ tio, ;; 11i), (:.-1), -',d '1 -2 ,

'I .!~''].1~-:1 ts tl-:~
illl 'r

*T 2 = Va
a2 sil? '


si n2 -

., l I + ( (:-7)

a s?.- i of sound in free streak

S r s t re am Mach nurb r


'i.,-: approxir nations L t constanlt aidl ".= :. ate
likewise adequate fr th prent p l7-. e; '. .for, as

.::n c 17

S'. -
\ L .


on t tion

I 2-- 2 / ,,
v + 2 0,2 "..'C. + + +. 1 + '1
(2 ? c10g 'i + '.C4 ._

2 lo+
.2 i + /1 -. + C.

xdhe r


Equat ... n (3B-0) rjn'o'vid1 -, thie des_ re.. r-e1 Pti n l;?tv.eeqn the
e f ,ctive .-,-h nu.-ber ?' an t'.e tr .---,r '"a ?h numn -r :.r
cr ae i- eq'j to..n ,'-_ A -'n .Af tA-,te vaerie ionn of
"" l)zC ", {
in figure l-,re t!_-t, ini sj t, of t:e r: -ld risE: of
i /". vi th ce2cressin '.' Un" fTr' itD-.--,t-c.jeed .ropueller
o;per nation Ie d. res e9 .

Tt, msy te nr, te-i t.:l- t e..u.t' Ls i'3- ) Ln (3B-,) nr-
parab. lic ,ro:"r.t ,i-ns t:, th. Clauert co':.sres .: bil ty
factor *.'- Laust ons 7-6) tc ( --') :t.,-,
ho.ever, en! -e ,ur.,t o tih const l.t of the o9ar .lic
reprs-r Ei't 3i r. Thue- the li.1; ty of t-.:. equat Ic .-
1i not re3tr-.icted- to th -, c -'e of V i t t'i t r f cf

with '"ach nutber 1 st foil..e"s th. ,mla-iert rel'it!on;
the equc.t )n-s -: rr- vae- li d for ay -'ar ir .t n thaft :,r, 'c -
sa-pr: i r .tedl t.he renio, of in-ere: t by a cIaraol,

A I ) C.
aL ytc

v'here A and 3 are cons tnts.

The com-res-eibil ty c o.e ti n oeases to apoly at
'Mach numbsrr s Ebo :e tne critical b'nch number for the


1. Lanchester, F. W.: The Flying-Machine from an
I~n.ineering Standpoint. Constable and Co., Ltd.
(Lor.. n), 1917.

2. E?'ris, R. G.: Forces on a Propeller Due to Sideslip.
R. & II. No. 427, British A.C.A., 1918.

3. Glauert, H.: The! Stability Derivatives of an Airscrew.
R. & M. N?. 642, British A.C.A., 1919.

4. :-lh-ert, H.: Airolane Proc.-ll. rs. Miscellaneous
Airscrew Problems. Vol. IV, div. L, sees. 5 and 6,
ch. XII of Aerodynamic Theory, '.. F. Durand, ed.,
Jul.i's Sprin. : (Perlin), 1935, pp, 351-359.

5. Goett, Harry J., and Pass, H. R.: Effect of Prop-ller
Cr._.i.tion on the Pitch.-in ;:icments of Sin-'-l-Engine
l'on:!planes. LiACA, A.C.R., 'ay 19'l.

6. Pistolesi, E.: J..w Considerations Concerning the
I-rjblem of Pr' clle's in y'L', (Trans.)*
L'Aeiotecnica, vol. 8, no. 3, March 192?, pp.177-192.

7. i."'ann, G., and W.inig, F.: Die Krafte ".ind K.'irr. nte
dter Luftschrm-I.-be bel Schr'''an'l:.zuns und FluA-zeui-
dr hurng. Luftfahrtforschung, 3d.. 15, Lfg. 4, Auril
6, 19 -'', pp. 206-213.

I. Bairstow, Leor.:rc:L A'plied Aerodyna..ycs. Lonimans,
Green and Co., -d a d., :;-w York, N. Y., 1939.

9. 1:isztal, Franz: The Problem of the P2'o.:peller in Yaw
with Snaeial R-ference to Airn1lan. Stability. T.M.
IT). C :, I:A A, 1933.

10. Rumph, L. B., White, R. J., and -rrummLn, H. R.:
Propeller Forces Du! to Yaw and Their Effect on
Airplane Et-:'.i]it,. Jour. Aero. Sci., vol. 9,
no. 12, Oct. 1942, pp. 465-470.

11. 'Weiss, Hcrbert ..: D;'-:..i'cs of Con-stant-Spced Propel-
lr:.. Jour. Aero. 3ci., vol. 10, no. 2, Feb. 1943,
P. 'o.
*Available for r'c, :rince cr lo-.n in Offire of Aeronautical
Intclliignce, National Advisory Comniitter for Aeronautics.

12. 'un':, Max F.: Fun'lamentals of Fluid Dynamics fcr Air-
c0aft D>ci-ner;s. The r.ons.alr Press Co., liew Yorh:,
N. Y., 1 ,9, pp. 15-23.

13. hibncr, i{erbrt S.: Formulas i'or Propellers in Ya2
and Char-ts of the Side--FErce: DerivEtive. i.Cn
ARR NIo. -E19, May 1943.

14. Rib.ner, Herbert S.: ProTosal for a Propoller Si-de-
Force Factor. NAC.t. RB io, 3LC2, Dec. 1943.

15. Bramwuell, F. H. R..Ti, S1 F., and sryant, L. W.
Exp.rim;rits onr l/oel P' at thae national
Phi;.ical Lp.orator.s-., (ti) E: prliernts to Detcrmine
th:e Lateral I'orc on a Pronr,:liecr In a Si'ewiirw .
R. & 14. Eo. 12,'3, Bitis; A... tC., I1914.

b1. Lesley, E. P., WorLvy, G3or;:e F., an.d 1-3v, Stanley:
Air Fro.elle.:5 in Yaw. Rer. I:o. 597, iACA, 1937.

17. Rncikel, J,-.c- F.: L.e tEfect c Pi.Xch on Force and
Moment,,-ter-ist.1cs (Jf 2,.;.-.cale -ropellers of
Five Solidiities. iCA A.h.T1., Ju.F 1942.

Figs. 1, 2

Figure .- Vector rel/of0iOns at a b/ode element

--. .. Resu/ldft wind
o f propel/er including
inNlow ondo sldewosh, :'V.4
Sxia/ velocity indc/d
-.n. .in. f../o.w =fl + d
-i --Side-vyind velocity ,/-
eluding side'wash,

Sd --'Cmponent of side wind
rrral to r (/ -)
Z s/in =-dVg

Fitre 2.- Vector re/a'ons fr crope/oer ir yaw.


Figs. 3,4

Ve/oc/tfy / outside
the s//pstreorn

A// air ve/oci/'es measured
re/ofi/'e to propeller hub.

Figure 3- Vector relo/ions pertaining o the
sidewash of a prope//er in yaw.


Fi ure 4.- Flow /;duced by the 5/1dev ise mo /Ion
of an infi'nte cylinder in a fluid ii tia.//y ao.


Figs. 5,8

Filcre S- Perspective view of three -do/iensiohal
graph of the assumed incremental J/ nf7o/

F comcorelnt of the f/ow in the p/ote of the
,rope//er ds/r.

__ ___
n~ t-t~-r rc"e)


N.- CA

Fig. 6




E- 4




ST n i i i

8 1- -e -. -
i 'i t t .

r : n t t i
t .

0 i 2 :i 4I

t.i,: l ...'i.,- :-r e not ,t 1 I .

Fi.:. 9






0 -'
5 -a



[t 0

;: .

Fi,. -0

I. I- ii
. l.... i- ..... I .. .I .. .. .. .. .. .. .
I. _... I i. 4.
.:L I
_ II

.. .- .. ...t ... .. r r.... -+._ .... ... ... .... .

..... j .-... .... .......^ .. -j-. .^ .-.-.. -. ...... ...... .
1 -... .. .. ..!. |. .... t .... .. .. J J
' 'I- I :i

....... .... !. ........ ...4 ---- .c.. -
~. i. k "

..: _. .. ..-..-. i L_ -.i_- ...:.
I ij \ i

I' \ i '
. .. 1
: I ;
1--- 1-- -----1-----.--- -ii*~ III
" T T "; r I

I .i. .\.i .. ," .i. ..
i I i i
.. .X .. ."I- I.. .. C.... ..

.- '.... ... '-- .. ... i ..... .. .....- ... ... J ... ",
I I I i

r~'^ "" [i -; T ';
. ... -- --- *---

i .j T .. I .
) i i

.. ._ _- _- **;- V _. -
T '\
[ _I i J ,,

*, i

.--. .-- --- j.
I '"
. __ .,___. ..___ ,, :~
-II- I '

- II11

7t 0
E- .
c '1


r-4 r

y '.
Fr o

O o
*i -i





7i.. 12







0 0

0 N

i1,:. 13

- t 7-, i .. .. rL a rstal .. 1 1
.... ..... _.-- ...... .. -...-.. .. ... .. .

11 0
1 -- .- ,. .6 .....-.. ...- --. ------
Y '\ + |' I I I

0 + L r. 's ... .7,:..,, f 1-j

__ ] L ..
i- -- .- .. .
!.. .-i-i -- --..-

.... i "I .... ......., r :.^ ^" I i t.. i. ...

i ,. __ 1, i, i *

S-li 1! i. r

:I n C :
=J.. t J_ 2. .., iJ.. 1 J.. -..... .. .. .. ... 1

E- 7or j:'- c ul *d:0c.iL* t 1 .1 -. .s .' ** :

~AFA i 14

i -r- Calculatd ----
,\' ). ,.' ':. :-o in: ntal -

\ \\ 1 -*_
..-. .. __ --- ....... ." -.- *, .- 'ld, d ....l--

3, -.' .---._ |

S. ,
..- ..- ..^ s .. i i .... -.

I v,: i i c z r- -
I I.

car ro. .for- -' i .. I 1 A
-.. 1 .. i -. .... ... .. .. ,. ,

i I -j .. .
J', _* ,- i ... .. ..a0. "- -- _
'3 .. ,I = 7 -\ ... i- .J I.-"

S'.. .. _.... .. ......- ... .........-

i 1,
.. .. "' -- :' L .. -

b ....-- -..
id- ax, ._ .- -
-. .. L ..... "-. .-... .. ...

. .i.-- !

to pitchi, v'ithi a:i::i .xu:, ha''.. b.-. co.:'.t',,. "itk v';.. ,-ct to, .: th
bo iD LL:A L.

NACA ig. 15


1 >
.-4 a

+ ... ... .4. --

rI 4

1 ,: 1 ';0

-I I ILr Iu -;.
i i I II -!
S,- r. I -' >
._ L--.-. L- -- ,

0 ,4*

.. -- ___ I -..... .. + .. .. .. -
i ^ ir'' !I I I : r '' o

I r"I I .c. .

i j .' ; |
i I
'p.-- ---- g -. 10

--i -r---"r -- r---- -Y- ^[~~~ *-^-r- .

!r ; *< I

l 111lll 11 1 1I
3 1262 08106 515 2


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