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crease in lift and drag coefficients. The port wing experiences opposite effects.As
a result of this imbalance in spanwise lift distribution, the wing develops a rolling
moment, which we will consider later while studying lateral stability. The imbal-
ance in drag forces gives rise to a yawing moment. For low subsonic speeds, we
can approximately estimate this yaYwing moment using the simple strip theory as
discussed in the following.
In the strip theory approach, the wing is divided into a number of spanwise
elements or strips. The aerodynamic forces on each strip are calculated assuming
that it is a part of a two-dimensional wing having an identical airfoil section as that
of the given strip. This concept is illustrated in Fig. 3.71. In other words, the strip
L.!
T
a) Strip RT on a firtute wing
b) Equivalent strip R'T' on a two-dimensional wing
Fig. 3.71 Concept: of strip theory.
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
theory ignores the downwash (induced angle of attack) variation along the span.
This amounts to ignor:ing the induced drag of the strip and an overestimation of
the sectional lift-curve slope. In view of this, the estim~tions based on strip theory
are at best only first approximations. Nevertheless, this approach is quite useful to
get an idea of the variaoles that can have significant influence on the aerodynamic
parameter ofinterest. .
Let c(y) be che local chord, ao(y) be the local sectional lift-curve slope, and V}
be the local resultant velocity of the strip RT on the right wing. We assume that
the angles ct, p, and r are small so that their products can be ignored. Here, we
resolve the sectional lift (dL) and drag (dD) forces along the stability axis system
because all the'aerodynamic (stability) derivatives like C,u3 are no::ally referred
to the stability axis system (see Chapter 4 for information on various axes systems
used in aircraft dynamics). Here, Oxy is the stability axis system with origin at
O. The z stability axis is not shown in Fig, 3.70. \Xf"th this understanding, the
component of force along Ox for the right wing strip RT of width dy (Fig. 3.70a
and 3.70c) is given by
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)
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