APPENDIX A
APPENDIX A
EQUATIONS OF THE THIN.AIRFOIL THEORY IDENTIFICATION OF PARAMETER8 The conversion of the equations for the aerodynamic 5_),.,t; etc. The subscripts indicate the variables characteristics of a finite airfoil based on the thin- held constant when the partial differential is taken.
airfoil theory (references 1, 2, and 3) from the old The equations now become British system of .aerodynamic coeiiicieuts to the stand- ard NACA form and the use of symbols for the param- eters, or slopes, in these equations has led to some mis- understanding as to the identity of these parameters.
The purpose of this analysis is to clarify the identity of \ tav,t /$t,_t \ _xt f /e.,,_ \tarot /c.At the parameters and to distinguish between the ones that are sometimes confused because of a similarity in eta t ova t _caf form. In addition, a summary of the relations is given whereby otber useful parameters not presented in fig- ures 1 and 2 may be computed from these data.
_t=/--_--/ a.+/--_-r-. / $_+1 .-':_z_ _, (4)
If These equations are of the same form as those pre-
C,, =/, ( ,_,_ I,_,)
sented in references 2, 3, and 5. By comparison it is it follows that possible to define the various constants of the equations in these references in terms of the variables involved.
('"v==_ _-I" _t _°tT _)_t t The following table of corresponding symbols has becn which is identical to prepared for future reference. The parameters from references 2 and 3 are, for obvious reasons, expressed
ac._ bC_ - bc.v
d C._r ---- "_"_a d a "t" _ T 7 d_ t q"-_-/ d_ , in tcrms of the old British system of coefficients; the angles were measured in radians; the pitching moment was measured about the airfoil nose.
NACA [
_-_ _ /
system !
Old British system of eve/fie;eats of coef- ficients Paralnctcr
Likewise if
Refer- Reference 2 Reference 3 ence 5
C..=/.. ( CN,_,,_,)
it follows that bCs f£1 i)a ]_,_
(oc_
and if _ a_ -- X, or -- Xt ¢l I
Then -- h. or -- X$
be., I 1 d _ _C_t _Cht bC_t
4 -T
'J,,= b-_.dC.,. + --_t d_ /+-_ d_,
_ be._
--m -- m, or -- mt \ _t/c,,,_
or, if it is considered that
i)c., -- e_o or -- _ -bt bC_t bC+_ 5C_t .
Because, according to the thin-airhfil theory, a linear -- H be_ ]_f,_, relationship exists among the variables C_, C_/, (-',., a, $_, and $,, the total differential in the foregoing equations -- b,, or -- bt,_ may be replaced by the variable. Because no change --b,, or --b_,_ in circulation is involved, (_-_._--- is identical with \ oot/ Ctt,Jt REPORT NO. 721--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS SUMMARY OF RELATIONSHIPS
The slopes summarized iu the following equations
are useful for design purposes and may be computed
with the aid of the charts of figures 1 and 2.
9a"
(gcv _:gcv
\ _: J.. _, \ 9a }5,, 9a 1_, ,, \ be./5,. _,\ aa/,,, _,
APPENDIX B
APPENDIX B DEVELOPMENT OF FORMULAS FOR TRIM, BALANCE, AND FREE-CONTROL CONDITIONS For an airfoil with a flap and a trimming tab, the formula for the tab deflection required to trim, where for trim Ch I is 0, was developed in the following manner.
From tile thin-airfoil theory (see appen(ILx A) C.v / SCv\ 5a ba 8
c,,= _v+(._7)., _,_/--=J _, (::)
,1, \ Oft J¢., af af, $1 S.lve for 3/in equation (1):
_,=_L k _- ),,,,, k _- J,,.,,t ,]<..,, J
Because Chs----0 to trim, equation (3) may be equated to 0. Solve for _tw%-01 and obtain 5e_t _'_t _,,_,,.o,=- (_,,) (._,, \ _)_I/¢., L, [f ('.N for the condition when C,¢_0 is substituted for Cs in equation (la), _¢ will become _¢,¢.h .0). Now equa- tions (la) and (3a) m_y be equated ant! "he res_.itant expression may be solved fi)t" St to trim _¢<c%-01.
(._"4 7
,, ,J ,, 5,<%.o,= L \_-_/,, .... ,. .,
(__") (_,_,) (_)
\¢'_I 1¢.,8, In this form, the tab detleetion to trim may be determined by direct substitution of tim wdm,s for the parameters as given in the data for this report.
The flap deflection with tiw tab s_,t to trim may be dctermim,d from equation (Is), which, when eonlbim,d and rewritten, becomes
"(<"°): (d_ /_-°'+(_')*'."_'(°"_) (0)
ki_j/e..L,I..k _)a iv,L, The equations for an airfoil lind a flap with a hahmcing tab were derived as h)llows: For a balancing tab, St isJOD, so that _,-----K_Iz+_, where K is a constant for a linear variatiou of _ with _z, and Se is the initial tab setting. Therefore equations (I) aml (3) become _C_A _ _ (Ka_+a,.)] .J and (_,:(_.,_,_, C.vt /_e,," I [_e,,,'_
t-*,-), 2' _t-_;),. ,, (r<,, _ ,,,,,_
\ _l/Lt,,fl ,, ., Wil,h controls fl've, Cht=O Itlid t,qllill,illlt (;i) lll!ellnil,,_ t, _', I_,,_, Revise equation (1) by changing ('_ to (?'_¢w_f-") and slibstitute _sie, s.,_ h)r _t; IlSe, this exprl,,_sion for ('.v(e_t.,,) in the foregoing relation, and the tlal)angD for control-free comlition b_,conms REPORT NO. 721--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS --- -- as+ - -- -- _, + "_, a, o
I
I ......t_.),,,,,t _°),,,, L t_.)',',t ...... _°),,,,t_ ,)_ ,, t ,)_ J 51,.e,,._ -_ /ac,;_ 15C_\ Ib-\ /b%\ _.J- 15%\ /bC_\ lba\ 15%\ 11 (7) . _ __ __ __ +-- + .... + --
[ t_g,)l,.ht()a);t.,._'r)e..,, t_l)¢..l, KL t_.),,,,,t_.J,,,,t_,)°.,, t_,)_.,,_
The equationfor the normal-force coefficient with free controls is obtained by substituting the free- floating flap deflection from equation (7) into equation (1). Thus C_ bC_ _a a, b- By the actual substitution of the right-hand member of equation (7), this eq.ation may be written as ba _)a _)a
k_.),,.,, t_),,.,.=.+L-t-a.),,.,, t_),,.,,t_,)<..,, +t_,),..,, 2'0
ia%'_ i_C_,\ ib,,\ 1_%\ +..I- IS%\ /bC,_\ 1_,_\ Is%\ (Sa) -t,-_.),,.,,t._),,.,,t._,),.,,+t_,,),..,. %-t._),,.,,t,_),,.,,t_),.,,+t_),..,, By the differentiation of equation (7) with respect to a, _f, o being a constant, the stabilizing factor becomes
(_q P%
=_ \ _,/a,._, \ _a/I,.I, If equation (8) is differentiated with respe(,t to a, the slope of the normal-force ('o(,ffieient em'vc I)(,com(-_ _C_ i_C_ ba ba _I (-_a"a)c'jli..-o = (_)s,.ll,{ 1 --[( _),.,tl, + K(_)e.,,,_ _)caj.-e} (1 O) or by differentiation of equation (8a)
,<()
, ,_.,,,,,._ _-/,,.,,_,/°..,.+i_),..,, +__-t_),,.,,_,,,.,, +t_,)o..,, By the use of the slope relations summarized at the end of appendix A, it can easily be shown that equa- -_J/"_' (Oa) (--_at)e,t-.=--[bO.t \ [_)C_,\ tions (5), (6), (7), and (9) may be considerably sim- (--_--7 ). , + hl ___ i plified. When this simplification has been made, these equatons read as follows: ,,_c,,, _,_ Its-,)...,, + :t_)_..,,/,,o_, 0"(%- 9 C_(%-o) +.
_k _,]..,, k_-7#_.,,,, \_,)o..,. (,_a) -- _t _ REFERENCES I. Glauert,H.: A Theory of Thin Aerofoils. R. & M. No. 910,.
BritishA. R. C., 1924.
c,,(%.0) ,_, +tf,(%.,,) 2. Glauert, H.: TheoreticalRelationshipsfor an Aerofoil with Hinged Flap. R. & M. No. 1095, BritishA. R. C., 1927.
3. Perring, W. G. A.: The Theoretical Relationships fol" an Acrofoil with a Multiply Hinged Flap System. R. & M.
/aC, A /bC, A
No. 1171,BritishA. R. C.,1928.
, _ _,_''-'_,,.,,°.+,-a...,2', 4. Si[vcrstein, Aim, and Katzoff, S.: Aerodynamic Charae- _'(",,'9 -- 75_',,\ _L/'_C_,'_ (7,0 tcri,tics of IIorizontal Tail S,rfaces. Rcp. No. 688, t,-_-;):.,,+'lt-a:):.,, NACA, 1940.
FROM NACA PLAIN-FLAP AND TAB DATA 17 CONTROL-SURFACE CHARACTERISTICS 5. Goett, Harry J., and Reeder, J. P.: Effects of Elevator Nose with an 80-Percent-Chord Plain Flap and Three Tabs.
T. Y. N'o. 761, NACA, 1940.
Shape, Gap, Balance, and Tabs on tile Aerodyn.o.mie Characteristics of a Horizontal Tail Siirface. Rep. N'o.
10. Harris, Thomas A.: Reduction of Hinge Moments of Air- 675, NACA, 1939.
plane Control Surfaces by Tabs. Rep. No. 528, NACA, 6. Silyerstein, Abe: Toward a Rational Method of Tail-Plane 1935.
Design. Jour. Aero, Sci., voJ. 6, no. 9, July 1939, pp.
11. Anderson, Raymond F.: Determination of the Charac- 361-369.
teristics of Tapered Wings. Rep. No. 572, NACA, 1936.
7. S_reet, William G., and Ames, Milton B., Jr.: Pressure- 12. Silverstein, Abe, and Katzoff, S.: Design Charts for Pre- Distribution Investigation of an N. A. C. A. 0009 Air- dicting Downwash Angles and Wake Characteristics foil with a 5{NPercent-Chord Plain Flap and Three Tabs.
Behind Plain and Flapped Wings. Rep. No. 648, NACA, T. N. No. 734, NACA, 1939.
1939.
._. __mcs, Milton B., Jr., _nd Sears, Richard [.: Pressure- Distribution Investigation of an N. A. C. A. 0009 Air- 13. Prandtl, L.: Induced Drag of Multiplanes. T.N. No. lg2, NACA, 1924.
foil with a 30-Percent-Chord Plain Flap and Three Tabs.
T. N. No. 759, NACA, 1940.
14. Weick, Fred E., and Jones, Robert T.: l_sum6 anti Analysis 9. Ames, Milton B., Jr., and Sears, Richard I.: Pressurc_ of N. A. C. A. Lateral Control Research. Rep. No. 605, Distrilmtion Investigation of an N. A. C. A. 0009 Airfoil N'ACA, 1937.
u, S, GQV(RNm_NT PNINTING OFFIC[:Ig41 # I I Z Positive directions of axes and ansies (forces and momenta) _ shown by _rrows I Axis Moment about axis Angie Veioclties Force I I(para]lei [ Designs,. Symo (tempo- Ang_xl Designation Designation S_- to axis) 8_- Politive tioa be{ direction nent along axis) r. Y---'-_Z Roll ..... _, X u p Longitudinal .... X [ Polling .....
M g-----*X Pitch .... $ q Lateral .......... Y N X--', l" Yaw ..... W r Normal .......... z Z Ya_nf ....
Y I Pttuhing ....
Absolute coe_cients of moment Angle of set of control surface (relative position), 5. (Indicatesurfaceby proper ._ C,-.q-_ .= M (roiliag) (pitcKing) 4. PROPELLER SYMBOLS D, Diameter C- P P, Power, absolute coei_cient e--_ p, Geometric pitch p/D, Pitch ratio C,, Speed-power coefficient _- _/p-_ . V', Inflow velocity ._, F_ciency V. Slipstream velocity n, Revolutions per second, r.p.s.
T, Thrust, absolute coemcient Cr--. ---_iz • , Effective helL, angle _ tan"(_) Q, Torque, absolute coefRcient Co--_---_ & NUMERICAL RELATIONS 1 hp.==76.04 kg-m/s=550 ft-lb./se¢. I 1b.--0.4536 kg.
I metric horsepower=l.0132 hp. t kg_-2.2046 lb.
I m.p.h.=0.4470 m.p.s. I mi.m 1,609.35 m_5,280 ft.
I m=,3.2808 ft.
! m.p.s.==2.2369 m.p.h.
O % ,r