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For an untwisted rectangular wing with a constant chord and an identical airfoil
section along the span, Eq. (4.574) reduces to the following form:
(CLp)W = -61(0o + CD) (4.575)
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410 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Usually, lao(Y)I > CD,I SO that the sign of Ciy depends on the sign ofao(Y). There-
fore, for angles of attack below stall where the lift-curve slope ao is positive,
(Ctp)W < O, i.e., the wing provides a positive damping or simply damping effect.
For angles of attack above the stall angle, the lift coefficient decreases with an
increase in the angle of attack so that ao becomes negatrve and (Ctp)W > 0. This
loss of damping in roll is one of the main causes of autorotati,on of the wings in
poststall region and aircraft spin entry, which we will discuss in Chapter 7.
The above result based on strip theory is very approximate because the str:ip the-
ory ignores the induced drag and mutualinterference effect between adjacent wing
sections. As a result, the strip theory prediction becomes increasingly erroneous
as the w7ing aspect ratio becomes smaller. For such configurations, the following
Datcod relation can be used:
(Clp)W=(P~ ),,=o(~)((C~C)) )/rad (4.576)
In Eq. (4.576), it is assumed that the angle of attack is in the linear range or
CL -. awa and the effect of drag force on the rolling moment is ignored. The
parameter (Cip)r/(Cip)r = o is given by the following relation:7
where
(Cip)r = (1 _ 2z'sin F +3zz sinl F)/rad (4.577)
(Czp)r =o
z' = 2z,y (4.578)
b
Here, zw is the vertical distance between the center of gravity and the wing root
chord, positive for center of gravity above the root chord.
The data to estimate (pqp/k)cL =o are presented in Fig. 4.25 for typical wing
planforms.
The vertical tail contribution, (CLp)V is given by
where
(Clp)V=l2(b)( b )fcyp,v
z - zv cos a - lv sina
(4.579)
(4.580)
Here, zy is the vertical distance between the aerodynamic center of the vertical tail
and the center of gravity and is measured perpendicular to the fuselage centerline,
lv is the horizontal distance between the aerodynamic center of the vertical tail
and the center of gravity and is measured parallel to the fuselage centerline. The
parameter C),p,v can be obtained using Eq. (4.547).
For supersonic speeds, no general method suitable for engineering purposes
is available for estimating the contributions of the wing and the vertical tail to
damping-in-roll derivative. Datcod presents data for some selected wing plan-
forms.lnterested readers may refer to Datcom]
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 411
Estimation of Cnp. This derivativeis a measure oftheyawing momentinduced
due to a roll rate experienced by the aircraft. The contributions of the fuselage and
horizontal t,ail to Cnp are usually small and can be ignored. The contr:ibutions
mainly come from the wing and the vertical tail so that
Cnp = (Cnp)W + (Cnp)-V
(4.581)
For low subsonic speeds, an approximate estimation of the wing conLribution
can be done using the strip theory as follows.
Consider once again the strip RT on right wing (Fig. 4.27a). The force in the
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