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For an untwisted rectangular wing with a constant chord and an identical aufoil
section along the span, Eq. (4.592) reduces to
(Cnp)W = -~(CL - CD")/rad (4.593)
Usually, for an airplane operating at an angle ofattack below the stall, CL > CDa
and the wing contributiori (Cnp)W iS negative.ln other words,if the aircraft has a
positive roll rate, then the wing develops an yawing moment that tends to yaw the
aircraft to the left.
It may once again be recalled that the strip theory ignores the induced drag
effects and the mutual interference between adjacent wing- sections. Hence, the
above strip theory prediction of (C,,p)W is quite approximate and will be in error
if the wing aspect ratio is small.
For more accurate estimations, the following formula can be used for subsonic
speeds:7
(Cnp)W=CLptana(K-1)+K(CC,)C=O.MC/rrdd (4.594)
where the parameter K is given by Eq. (4.549) and
~C)CL_O.M=(AAB++44 AA~)[Ajj::::~AB+ A) AA ]
5(A + cos Ac/4)L
x(CC,),.=o7rad . (4.595)
where
B = \lL M2 COS2 Ac/4 (4.596)
A+6(A+cosAcl4)(~~_/+ ~ )i
:--- )] (4.597)
(CC,)..=o=- 6(A+4cosAc/4) J
Here, A is the exposed aspect ratio (At) and g is defined in Eq. (4.490).
The vertical tail contribution is given by
(Cnp)V=-(/j;)(l.,osa+zusi.)( b )Cyp.v/rad (4.598)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 413
Estimanon of Cyr. This derivative is a measure of the side force induced due
to yaw rate experienced by the aircraft. Generally, the contributions of the wing,
fuselage, and horizontal tail quite small and can be ignored. The only mearungful
contribution comes from vertical tail, which can be estimated using the following
formula]
(Cyr)V = -; (/,,os tY + zu sin a)Cyp,v]rad (4.599)
where Cy)S,V is given by Eq. (4.547).
For szupersonic speeds, no general method is available for the estimation of
(Cyr)V.7
Estimation of Ck. This derrvative is a measure of the rolling momentinduced
due to yaw rate experienced by the aircraft. Generally, the contributions ofthe fuse-
lage and the horizontal tail surfaces are small and can be ignored. The contribution
mainly comes from the wing and vertical tail surfaces so that
Clr = (Clr)W +(Clr)V
(4.600)
For approximate purposes, we can use the strip theory to estimate the wing con-
tribution atlow subsonic speeds as follows.
Because of yaw rate, the relative velocity experienced by the wing sections
varies along the span. Suppose the aircraft is experiencing a positive yaw rate.
Then the sections on the right wing experience a decrease in the relative velocitjr,
whereas those on the left wing experience an increase in the relative velocity. As a
result, the lift on right wingis smaller compared to that on the left wing. This gives
rise to a positive rolling moment. Thus, conceptually we see that (Clr)W > O.
The relative velocity experienced by the elemental strip RT (Fig. 4.27b) on the
right wing is given by
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