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or, in coe:fficient form,
and
2pr /2
(Cl)r.w = __ Spbr l,'/2(C _ CDa.j)c(y)ydy (3.268)
2F '2
(Cnp)r.w = ~~- [b'2(CL _ CDa.l)c(y)y dy (3.269)
For a rectangular wing with a constant chord c, the above expression reduces to
F(CL - CDcr.l)
(C p)r.w = --
4
(3.270)
where Ct.L iS the local (sectional) lift coefficient of the left wing.
The net or total yawing moment is the sum of the yawing moments due to the
right and left wings and is given by
266 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
The simple strip theory analysis, even though very approximate, has given us
important information that the wing dihedral has a destabilizing effect on direc-
tional stability and this effect is small at Iow angles of attack but may become
significant at high angles of attack or high lift coefficients.
It may be recalled that the strip theory ignores the induced drag effects; hence its
predictions will be increasingly jYn error as the wing aspect ratio decreases. For such
cases, the following empiricaIlformula9 may be used for preliminar)r estimations
at low subsonic speeds:
(Cnp)r.w = -0.075 FCL/rad
(3.271)
where the dihedral angle r is in radians. For supersonic speeds, no general method
is available for estimation of the wing contribution to directional stability due to
dihedral effect.i According to Datcom,l this contribution is generally small and
can be ignored.
Ejfect of swaep. The wing sweep-back has a stabilizing effect on static direc-
tional stability. To understand this,let us consider a swept-back wing of sufficiently
high aspect ratio and with zero dihedral operating at an angle of attack.c! and
sideslip p and moving at a forward velocity Vo as shown in Fig. 3.72. The velocity
components in the spanwise, chordwise, and normal directions are given by
vs - Vo(sin A cosa + p cos A)
Vc -. Vo(cos A cos ct + ~3 sin A)
-. Vo cos A(cos a -1: p tan A)
VN - Vo Sill CL
(3.272)
(3.273)
(3.274)
(3.275)
where the first (upper) sign refers to the right wing and the second (lower) sign
refers to the left wing when the wing is in positive sideslip.
As before, let us assume that both cr and p are small so that higher order terms
involving these two parameters can be ignored. With these assumptions, we have
Vs - Vo cos A(tan A + p)
Vc - 'Vo cos A(l 4: t3 tan A)
VN -. lVoa
(3.276)
(3.277)
(3.278)
As said before, the spanwise component of velocity does not affect the pressure
distribution and hence is ignored in the following calculations. It only adds to skin
friction.
The angle of attack of right (starboard) and left (port) wings are given by
VN
ai = tanat = yc
ct
cos A(l i- p tan A)
- a sec A(l +- p tan A)
(3.279)
(3.280)
(3.281)
Voc
Vo P
STATIC STABILITY AND CONTROL
A
h
r;
yh
267
v, ~-k ~ /::s-OCL
┏━━━━━┳━━━━━┓
┃ ┃ ┃
┗━━━━━┻━━━━━┛
Fig. 3.72 Strip theory analysis of wing sweep effect.
For le'vel fiight with zero sideslip, the local angle of attack is given by
so that
a's - cr sec A
as - ai = +a sec Ap tan A
(3.282)
(3.283)
The dynamic pressures experienced by the right and left wings are given by
qi - 21p(\/c2 + LrF~)
= 2lp Vo2 COS2 A(l i: p tan A)2
(3.284)
(3.285)
Here, we have ignored higher terms such as Ct2, avp, or [112. Forlow subsonic speeds,
we can use the strip theory to estimate the wing contribution to directional stability
due to sweep-back as follows.l0
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