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Differentiating Eq. (4.535) with respect to a and simplifying,
(Crrur)t
-2a, V,,7,(: ) (1:)
(4.537)
For airplane configurations wich long fuselages and short aspect ratio wings,
the contribution of the wing-body combination can be quite sigruficant. For such
configurations, the total value of Cma for the complete airplane is given by
Cma : (Cmd)WB + (Cmd)t (4.538)
The wing-body contribution can bc estimated using the following expression/
(Cma)WB = [KWcB)+KB,W,j(SS )(C.,).
+(Cm*)BSS,2 7,ad (4.539)
where (Cn a)e iS the contribution of the exposed wing and (Cma)B iS that of the
isolated body.
For both subsonic and supersonic speeds]
(Cma)e = (C:na.)e+ (-: )(C,,)e (4.540)
where Xcg.U is the distance of the center of gravity from the leading edge of the
exposed wing, positive aft (see the insert at the top of Fig. 4.2(}), and (CLa)e iS to
be evaluated as discussed earlier.
For subsonic speeds,
(C:d)t = - ;:;) (-., )j(C,tr)e +;C_(g) (4.541)
The function Cmo(9) can obtained using the following expression, whichis a curve
fit to Datcom datal
Cmo(9) = ( p )(0.0008 r4 _ 0.0075 r3 +0.0185 r2 +0.0128 -c - 0.0003)
(4.542)
where r = pAe, Ae is the exposed wing aspect ratio, and p = JC~7.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 403
(C:ia)e
taper ratio=0.25
c Crna)e
- A -.0
.- A = 30o
-.. A - 45o
.. A -. 600
taper rat1o=0.5
Aspect Rat1o-.2.0
Aspect Ratia=4.0
Aspect Rati0=6.0
Fig.4.23 The parameter (C:a)e for subsonic speeds:r
For supersonic speeds] the procedure of evaluating (C:rtrcr-)e is quite involved
and is not discussed here.lnstead, using Datcod data, we have performed some
calculations and presented the data on (C:a)e for some typical wing planforms in
Fig. 4.23. As noted in the case of ccZcr.)e, the wing leading-edge sweep and wing
aspect ratio have more significantinfluence, and the wing taper ratio has very little
effect on (C:na)e.
The parameter (Cmcr)B can be evaluated as follows]
For both subsonic and supersonic speeds,
(Cmcr)B=2(C;ncr)B l.:,-,.V,,](S V,.,./,) (4.543)
where xci, Xm~, and VB1 are defined in Eqs. (4.513) and (4.514). The parameter
(C:a)B can be determined using Eqs. (4.515) and (4.517).
ECtUATtONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 405
--- 7L~O
---- x=o.5
-.-.. x=i.o
c J!
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