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Ox direction is given by
dF - dL sin Cep - dD cos ap
where
(4.582)
= ~:pUo2[ao(y)(a + ap)ap - (CDO,l + CDa,l(cy +ap))lc(y)dy (4.583)
= ;: p Uo2[_CD.I + (ao(y)ct - CDa.I)CX plc(y) dy
CD.1 = CDO,I + CDa,.lCL
(4.584)
(4.585)
Substituting ap = py/Uo, the yawing moment developed by the elemental strip
RT is given by
dYM=-gpUo2[-CD.l+[ao(y)a-CDa.,lZrYlc(y)ydy (4.586)
The total yawing moment due to the right wing is given by
YMR = _: pUo2l,' [-CD.I + [ao(y)a - CDcr,L]7jY].(y)ydy (4.587)
Similarly, the yawing moment developed by the left wing (change +y to -y) is
given by
YM, = -,pU,~[,' [-CDJ - [ao(y)a - CD j]ZY]c(y)ydy (4.588)
The total or net yawing moment, which is the sum of the right and left wing yawing
moments, is given by
We have
YM=-pUo2['[ao(y)a-CD..,]GjY),(y)ydy (4.589)
(4.590)
(4.591)
Cn= pYUMS7
Cnp = -,(:C:: )
412 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
so that
-4 z
(C"p)W = Sb4 f,b/2Lao(Y)a _ CDa.j]c(y)y2dy (4.592)
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)
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