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aaM)f = 2(k2 -ki)qvf
(3.5)
where Vf is volume of the fuselage and (k2 - ki) is the apparent mass constant,
which depends on body fineness ratio (lf /b f.max) as shown in Fig. 3.6. Assuming
that the fuselage is a streamlined body with varying width/diameter, we can rewrite
Eq. (3.5) in a coefficient form as
k2 - ki)
(aaC.f=~k2s [,llbfdx (3.6)
M/
Fig.3.5 Ideallhud flow over an airplane fuselage.
Cmo > 0
dC <O
dty
170 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
┏━┳━┳━┳━┳━━━┳━━┳━┳━┳━┳━┓
┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃
┃ ┃ ┃ ┃ ┃ ┃ ┣━╋━╋━╋━┫
┃ ┃ ┃ ┃ ┃/--- ┃--- ┃ ┃ ┃ ┃ ┃
┣━╋━╋━╋━╋━━━╋━━╋━╋━╋━╋━┫
┃ ┃ ┃7 ┃/ ┃ ┃ ┃ ┃ ┃ ┃ ┃
┣━╋━╋━╋━╋━━━╋━━╋━╋━╋━╋━┫
┃ ┃ ┃( ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃
┣━╋━╋━╋━╋━━━╋━━╋━╋━╋━╋━┫
┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃
┗━┻━┻━┻━┻━━━┻━━┻━┻━┻━┻━┛
FINENESS RATIO
Fig.3.6 Fuselage apparentmass coefficient.!
where bf is the local width/diameter, lf is the fuselage length, S is the reference
(wing) area, and c is the reference length (wing mean aerodynamic chord). Note
that the values of slope of the pitching moment and that of the pitching-moment
coefficient given by Eqs. (3.5) and (3.6)'are per radian.
For an isolated fuselage (no wing), the local angle of attack ct f would be a
constant along the entire Iength and equal to a. However, in presence of the wing,
the local fuselage angle of attack varies as shown in Fig. 3.7 with upwash in front
of the wing leading edge and downwash behind the wing trailing edge. For that
part ofthe fuselage extending from the wingleading edge to the wing trailing edge,
the local flow is essentially parallel to the wing chord so that a f = 0. To account
'for these induced effects, Multhopp3 modified Munk's theory as discussed in the
following. Note that this theory is applicable only for low subsonic speeds.
According to Multhopp,
(ac )f = 2S [,U bj(} + aa )dx
(3.7)
Fig.3.7 Schematic diagram of the ffiselage flow field in the presence of the wing.
STATIC STABILITY AND CONTROL
┏━━━━┳━━┳━━┳━┳━┳━┳━┳━┓
┃,,,, ~ ┃- ┃ ┃ ┃ ┃ ┃ ┃ ┃
┃ ┃ 3 ┃ 4 ┃5 ┃6 ┃7 ┃8 ┃9 ┃
┃t -l~: ┃-- ┃ ┃ ┃ ┃ ┃ ┃ ┃
┣━━━━╋━━┻━━┻━┻━┻━┻━┻━┫
┃ ┃_- ┃
┗━━━━┻━━━━━━━━━━━━━━━┛
171
Fig. 3.8 Definition of fuselage nose droop and aft upsweep.l
where eu denotesinduced upwash or downwash at the axiallocation x. The slope of
the pitching-moment coefficient given by Eq. (3.7) is per radian. The zero~lift pitch-
ing moment Cmo. f can be assumed to be equal to zero for uncambered (symmetric)
fuselages. However, for cambered fuselages such as those with leading-edge droop
or aft upsweep, Cmo. f iS nonzero and can be estimated as follows:l
Cno.f = J;6 SSk [,"/ b?j(a,ow + icl.B)dx
(3.8)
where aow is the wing zero-lift angle relative to the fuselage reference line and /u. .B
is the incidence angle of the fuselage camberline relative to the fuselage reference
line. The parameter icl.B iS assumed to be negative for nose droop or aft upsweep
as shown in Fig. 3.8. Note that in Eq. (3.8), both aow and ict.B are in degrees.
Thus,in general, the pitching-moment coefficient of a fuselage can be expressed
Cm.f = Cmo.f + aac. )fa
(3.9)
nie v~rriation of a6u/aa for sections ahead of the wing leading edge is shown
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