APPENDIX A
l ’ NACA!t!echnical Note No. 796 n ’ APPENDIX A EQUATIONS OF THE !IHIN-AIRNOIL THXORY Identification of Parameters “ The conversionof the equationsfor the aerodynamic characteristics of a finite airfoil based on the tha thin- airfoil theory (reference1, 2, and 3) from the old British system of aerodynamiccoefficientsto the standard NACA form and the use of symbols for the parameters,or slopes, in these equationshas led to some misunderstand- ing as to the identityof these parameters, The purpose of this analysis‘is, to clarify the identity’ of the param- eters and.to distinguishbetween the ,onesthat are some- In addi- times confusedbecause of a similarityin form, tion, a summary of the relations is given whereby other .
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useful parametersnot presented in figures 1 and 2 may be computed from these data.
.
.
If = fl(ct,bftst) CN it follows that which i6 identical to: l .
-.-” . .
. . .
..” ,.
.
~, Likewise if it follows that a-c ac “ . acre” ‘“m cm =GdcN+— — dt3t d~f + ~bt a6f acN ,. -, .. . .
and if ,., .,. . . . .
.
,“’ ,:. : ‘“. e&= , f@N,~f, @ ‘ ,:, ,- .
Then .. .
— “’~Chf 5Ch:f. achf d~f + - det dchf = ‘dCN+— a8t a6f S.
aCN or, if it is consideredthat .
.
.
.
— Because, according to the thin-airfoi.l ”theory,a linear Chf, Cm, - relationshipexists’ timong the variables.’CL~, . . .
., ‘~t”,’ the total differential-inthe foregoing a, “8f’,and equationsmay be replaced by the variabl~. Because no -.
~a is i-den- ‘“ - change in circulationis involved, —
(as,)c
N,~t The subscriptsindicate tical with — s etc.
(::) f cn,8t the variablesheld constantwhen the partial differential is taken.. The equations now become cN=(*),f,,t~a-(+~n,, t~f-(*)c,,:t] ‘1) ,.
n_ , — * .
?’ 30 EMC.L Tkc-h3il’cal ‘No”t e.No. ?96 8*
chf = (%),,,,,, c“ ‘, G30,n,,. .’f+ (-).n,”,f~~ “(3)
., u.
achf ,/ ~hf a~ + Chf = ~aa ~f,8t
) () - “’f+ (-)a ,f” ‘4)
a6f a,6t These equatiousare of the, s,ame form as those pre- sented in references2,:” 3,;and 5.” By comparisonit is possible to define the various constantsof the equationsIn these referencesin terms of the variables involved. In order to indicate from which relationshipsthe various partial derlvat’ives are de’t’ermi”ned, subscr;fpts” are added to the derivativesto indicate the variablesheld constant in taking the partial derivative.
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The followingtable of correspondingsymbols has .
been prepared for futurereference. .The parametersfrom references2 and 3 are, for obvious reasons, expressed .
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in terms of the old British system of coefficients;the angles were measur’ed iti, “radians ; the pitchingmoment was measured about the airfoil nose.
,, NiiCA. syst,.eti”.of~ .,,Old Brit.i,sh” system .
of coefficients coefficients !
Parame,te’r ,..,, . .
Ref.,erence .5 ‘ Re,fe.re.nc,e, 2- Reference 3 . .
., I I ,,.
.,.
.acm “
.. . . ,,,. .,, .:-’, . . . .
al ..,. al al “t
(–)
act 6f,5=t”””
., .,..,.. .,,..
. .
. . . . . ,, ., I . . .
-.,.
.,.--- - ~q ,- , .---.?-%y, , ..,. “.:!.
I
I l -ha or -Ax “-.2..
, . .
.
.
t .
aa ---- -As or -ha
(–) 36
,, .
t cn,8f . . .
..
— NACA TechnicalNote No. ’796 31 ,.
.- -.
NACA system of Old B,ritish system of coefficients coefficients “, Parameter Reference 3 Reference 5 Reference 2 acm _l -— ---- z (–) 4 ac~ tif,8~ . . .
acm ,’ ---- -m .-m’s or -m1 ~ () cn,~t ac~ ---- ---- -m~ or -m2 (–) aat Cn,af ---- ---- -bl (achf) aa 6f,8t ---- ---- -ba aaf (achf) a,~t aChf bl _— -u ‘Br al () ~c~ sf,a~ bzal-blaa = -b -brr or -b -VII 1,1 (%) al f Cn,8t :, ---- T lg -brs or -bl,a (achf) a~t Cn,bf u. — . Summary of Relationships * The slopes summarized in the following equations are useful for design purposes and may be computed with the aid b of the charts of figures 1 and 2. ‘ ..
, ,, $’ NACA,TschnicalNote No.”796 *“ acN
() =- (%9 (~)cn,8f
~
6f,8t a,&f .
.
(%2)3,,,, “
(a%?8f,;t 6%!!,,,, .
.
(achf) — a,,,= -(%38f,8t (srj)cn,8, ‘(%)cn,,t a6f
(%) = - (%?),,,,, (~)cn,af + (>)cn,af
a8t a,&f
t
acm
ac
()
= &f,8t
=
.
(3+
acw
CN sf,at
()
=% f,8% .
?lCm .
.
~a,8t .
() = -(%),f,,t’(~)cn,,t ‘+-(* )cn,,t “- “-
acn = - (%9 (~) +(~)cn,,f (–) d8t u,8f t3f,8t cn,8f NACA TechnicalNote No. 796 =- (-).,5,
(“h’)
a6f a,~t
.
?
APPENDIX B
,,, NACA TechnicalNote. No, 796 APPENDIX B DEVELOPMENT OF FORMULAS FOR TRIM, BALANCE, . .
AND FREE~,.00NTROL CONDITIONS For an airfoil”wi~ha flap and a trimming tab, the formula for the ta% deflectionrequired to trim, where for trim Ch is’g~, was developed in the followingman- f ner.
, ,.
. . . .
From thethin-airfoil theory (see appendix A)
‘N= (af,,t~. -(%)=,,6, ‘f “ (%)cn,,f 8’] ‘1)
.
.
~chf~ .
Chf = .
( ‘N +(-)cn,,t ‘f + (%$)cn,,,~t ‘3)
acn ‘8f,&i Solve for ~f in equation (l): 1- acN
()
c~ - =J6f,6t
aa+ (.)%,6, (%2+6t -
~- (la) w
6f=- (2),f,6t (~)cn,,, Because to trim, equation (3) may be equat- Chf = O ed to 0, Solve for 8f(Chf = ~, and obtain If c~ for is substitutedfor Ohf = O c~ in equation NACA. TechnicqlNote No. 796 ,.
Now equations (la) will become
(h). sf
af(ch~=0)0 l and (3a) maybe equated and,tie resultant expressionmay be solved for 6% to trim “6t(Chf=O)s ( ‘“hf)
acn
1 cf}~t+ cN(Chf=@ ‘,facl~~ ~ chf~ (-) - (*: ~ [_ +4
=o)-
%chf
ikt
y
(--9
()
a% cn,6f
cnt~f - In this form the tab deflection tc trim may be deter- mined by direct substitutionof the values for the parameters . as given In the data for this report.
* The flap deflectionwith the tab set to trim may be determinedfrom equation (la), which, when com%ined and rewritten,becomes The equations for an airfoil and a flap with a balanc- ing tab were derived as follows: For a balancing tab, 6~ is so that f(af), 6t =
Kaf + ate, where K is a constant for a linear
.
variation of 6t with and 6*0 is the ini- ~fo tial tab ~etting. Therefore equations (1) and (3) become CN
=Caf,+a - (%)cn,,t ‘f - (*)cn,,:”f+ ~to)
L .
and 36 NAC~ TechnicalNote No. 796 Whf ~ ‘ach’f’ ‘
Ohf = —
‘( ) ‘f”+ (*)n,,f’”f + “to)
()
CN+. a~f cn,6t___ ,_ ..: aeti&f#at .
— ‘With controls free, Chf=O, and equation (3) becomes . .. .. ... .
achf ?@f
:&hf
—,
c~+~ -!- +6J
K~f(chf=o)
()
() ()
,~t*f(chf=o) ~c
f&h
“n 6f,’t3*’ ( )
n,8f , ..- Revlse equation.(l)%y changing CN to cN(chf=o) and substitute for tif; use this ex- bf(chf=o) .
“ pression for c~ in the foregoing relation, (Ch@=())” “ A .
* and the flap angle for orntrol-freeconditionbe- .“.
comes - . ,., .
. .
,, ,,, .
,.
.
,’ 1’ ,.
,.
., I .
. .
.
. .’ . “, * , f (O’hf=o) =- (7) ,J .. .
The equation for the normal-forrm coefficient with free controls is obtained by substituting c the free-floating flap deflection from eption (7) into equation (1).
i.!: Thus :. .
%~=o)f%),f,% [~”(%)n,,p(%p) -(%)%,8f@’f(~=or’t.)] ‘8)’
..
-4 .
, 1.
,%, ,, of equation (7), this equation maybe By the actual substitution of’ the right-hand menber written-as ,.
cd -4 ,, ,, .
,,. ~ ., ,“ .
,.
;, I ,,,, ,,, ,., !1, II 1.
.
(%) ~y ‘Ghe .&iff erentiation of equ.at ion (7) with reqect to CL, 6 ~ being a const=t, the .. “’. - Stabilizing factor “becajes ,, I =- .
.
.
,.
,’ .
‘.
* . .
l , If equation (8) is differentiated tith respect to CJ, the slope of the norml.-force coef- ficient curve becomes (lo) J = or by differentiation of equation (&) * ~ ,~ w m (lOa) I I I ,1 I I ,, ,., i ~, II, il. i. ,, ,,, 1,6 II I I 40 NACA TechnicalNote No, 796 By the use of the slope relationssummarizedat the end of appendixA, it can easily be shown that equations (5), (6), (7), and (9) may be considerablysimplified.
when this simplification has been made, these equations read as follows:
cN(c@))
cN(chf=@
aa -L (5a) .
cN(chf=@
‘t
.
(Chf=o) .
aa
-+ (6a) acN n .
() G a,6t achf
bchf
6tc Ua +
() () ~
a,tif
aa t3f,6t
a- (78) ,.
af(chf=o)
(achf) + ‘(y)a,,f “
a8f a,6t (9a) .
.
— + K (*)cn,,f]
(achf)
a~f Cn,dt (lOb) ,
.—
“+’(-)a,,f]
cc”)
a~f a,i3t
-.
.i NAQA Technic al.Note, +~o. ,796 b, REFERENCES, ,,. . .
,.
. .
— 1. Glauert,H.: A Theory of Thin Aerofoils. R_& M; NO.
910, British A.R.C,, 1924, TheoreticalRelationshipsfor an Aero- .j2.Glauert,H:: foil with Hinged Flap, R.& M. No. 1095, British A,R.C., 1927,, The TheoreticalRelationshipsfor b 3. Perring,W. G. A.: an Aerofoil with a Multiply Hinged Flap System.
R,& M. No, 11~1, British A.R,C,, 1928* 4. Silverstein,Abe, and Katzo.ff, S.: “Aerodynamic Charac- teristicsof Horizontal Tail Surfaces.
~ep, No.
688, NACA, 1940.
5. Goett, Harry J., and Reeder, J. P.: Effects of Ele- .
vator Nose Shape, Gap, Balance, and Tab8 on the * AerodynamicCharacteristicsof a HorizontalTail Surface.
Rep. No. 675, NACA, 1939, .
.
6. Silverstein,Abe: Toward a Rational Method of Tail- plane Design. Jour. Aero. Scl., vol. 6, no. 9, July 1939, pp. 361-69.
?. Street , William G,, and Ames, Milton B., Jr,: Pressure- DistributionInvestigationof an N.A.C,A. 0009 Air- foil with a 50-Percent-Chord Plain Flap and Three Tabs. T.Ii. No. 734, NACA, 1939.
8. Ames, Milton B,, Jr,, and Sears, Richard I.: Pressure- DistributionInvestigationof an N.A.C.A. 0009 Air- foil with a 30-Percent-Chord Plain Flap and Three Tabs. T.N. No. 759, NACA, 1940.
9. Ames, Milton Be, Jr,, and Sears, Richard I.: Pressure- DistributionInvestigationof an N.A.C.A. 0009 Air- foil with an 80-percent-Chord Plain Flap and Three .
Tabs. T.N. No. 761, NACA, 1940.
10. Harris, Thomas A.: Reduction of Hinge Moments of Air- plane Control Surfaces by Tabs.
Rep. No. 528, NACA, 1935.
b 11. Anderson, Raymond F,: Determinationof the Character- istics of Tapered Wings.
Rep. No* 572, NACA, 1936, l * NACA TechnioalNote No. 796 l “ Design Charts for stein, Abe, and Katzoff: S.: 12. Silver PredictingDownwashAngles and Wake Characteristics ~Os .648, Rep., behind Plain and Flapped Wings.
NACA,’ 19390 ,.
13. Prandtl, L.: InducedDrag of Multiplanes. T.N. No.
,.
. .
NACA, 1924. .
1.82, and 14. Weick$ Fred E., and Jones, Robert ‘1.: R6stim& Lateral Control Research.
Analysis of N.A.C.A.
Rep. No.”605, NACA, 1937.
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Di$i
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F*M :.- Vlriltion of (%,46)% , (bc#4n)6 , d with Of/o OF o~n for tb LImmm %/W,,,U .
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I , * l .’ l -m m ,, I ,, i!’1 ,, ,,, , 18,, , ;,l ,’, ,.
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Ui : :: +-’’,”+/ i
,, Ziv” ~~ , ‘, . ..— ..-- .--. — ------- ---.. ---—–-/k6L ~ ‘“ !, ,, chnf /mWo, f2V ,&hA A&m, 7.- W dce’ :)! .Il+jicol air!o%h+ di91riZw$an unff &# .
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