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dF - dL sin(at - a) - dD cos(aL - a)
dL(ai - a) - dD
With a't - a = pr', we have
dF - dLBr - dD
= ~p Vo2c(y) dy(CI.RPr - CDL.R)
(3.251)
(3.252)
(3.253)
(3.254)
where Ct,R iS the local lift coefficient of the right wing.
In strip theory, the sectional lift and drag coefficients are assumed to be given
by
Cl - aocti
CD - CDO.I + CDa.ICt
(3.255)
(3.256)
where ao is the sectional (two-dimensional) lift curve slope, CDO.t iS the sectional
zero-lift drag coefficient, and CDa,.l iS the increase in sectional drag coefficient per
unit increase in angle of attack. Note that this increase in sectional drag coefficient
with angle of attack above CDO,L iS caused by an increase in the profile drag
coeffic:esnt and not caused by the indtrced drag coefficient. As said earlier, strip
theory ignores induced drag,
Then,
dF = ~p'Vo2c(y)dy[CI.Rpr - CDO., - CDa.,(a + pr)] (3.2s7)
The yawing moment due to the strip RT on the right wing is given by
dN = -ydF
(3.258)
= ~p Vo2[_CI.Rpr + CDO.I + CD".l(a + pr)lc(y)ydy (3.259)
STATIC STABIUTY AND CONTROL
7~
The yawing moment caused by the right (starboard) wing is given by .
b/2
NR=~pVo2[_CI.RpF+CDO.I+CDcr.IGy+[3r)] c(y)ydy (3.260)
Similarly, the yawing moment caused by the left (port) wing is given by
N, = ~pvo2[_cL. f/r _ CDO.l - CDa.l(cr - pF)] [b'2c(y)ydy (3.261)
N=~pVo2[_pr(cl.R+Ct.L)+2CDa.IPf-J[b/2c(y)ydy (3.262)
We have
so that
Then,
CL.lR = ao(a + pF)
CLn = ao(r:y - Br)
CLJR + CL.IL - 2aoa
- 2CL
(3.263)
(3.264)
(3.265)
(3.266)
N=~p\/o2f/~ bl2(C _
- CDa.l)c(y)y dy (3.267)
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
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