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Wing 111
denotes outboard edge, subscript I denotes inboard edge), or the unused fixed panel span ratio gives the
equation (c0 + c1b/2)O − (c0 + c1b/2)I = fpb/2, which can be solved for the semispan b/2.
To complete the definition of the geometry, one of the following quantities is specified for each
panel: panel area Sp; ratio of panel area to wing area, fs = Sp/S; panel mean chord cp; ratio of panel
mean chord to wing mean chord, fc = cp/c; chord ratios λI = cI/cref and λO = cO/cref (taper); or free.
The total wing area equals the sum of all panel areas:
S = Sp + S fs +2 bpcp +2c bpfc +2cref
bp
1
2 (λI + λO)
If there is one or more taper specification (and no free), then cref is calculated from this equation for S,
and the mean chord is cp = 1
2 (cI + cO) =cref 1
2 (λI + λO), Sp =2bpcp. If there is one (and only one) free
specification, then Sp is calculated from this equation for S, and the mean chord is cp = Sp/(2bp), with
cI =2cp/(1 + λO/λI), cO =2cp − cI.
Since the panels have linear taper (c = crefλ), the mean aerodynamic chord is
S¯cA =
S =
b/2
−b/2
b/2
−b/2
c 2 1
dy = b
c
0
2 refλ 2 dη
= b c 2 1
ref
3 (λ2I + λIλO + λ 2 O)Δηp = 1
3 (c2I + cIcO + c 2 O)2bp
1
cdy = b crefλdη
0
= b 1
cref
2 (λI + λO)Δηp = 1
2 (cI + cO)2bp
These expressions are evaluated from panel cI and cO, as calculated using λI and λO, or evaluated using
the ratio λO/λI (cref may not be the same for all panels).
The mean aerodynamic center is the point where there is zero moment due to lift: ¯xACLS =
¯xA cℓcdy = xcℓcdy, with ccℓ = ℓ(y) the spanwise lift distribution. Thus
1
ℓ(η)(¯xA − xAC(η)) dη =0
0
The locus of section aerodynamic centers xAC is described by the panel sweep Λp and the offset xIp at
the inboard end of the panel. These offsets can be a fixed input, a fraction of the root chord, or a fraction
of the panel inboard chord. Assuming elliptical loading (ℓ = 1 − η 2 )gives
π
4 ¯xA =
1
ℓ(η)xACdη =
1 − η2 xIp + b
2 tan Λp
η dη
0
=
xIp
1
2
η 1 − η 2 + sin −1 η
− b 1
tan Λp
2 3 (1 − η2 ) 3/2
ηO ηI
where xIp = p
q=2
xIq +(b/2) tan Λq−1(ηO(q−1) − ηI(q−1)) − (b/2) tan ΛpηIp. The vertical position of
the mean aerodynamic center is obtained in a similar fashion, from panel dihedral δp and offset zIp at
the inboard edge of the panel. Assuming uniform loading (ℓ =1)gives
1
¯zA =
0
zACdη =
zIp + b
2 tan δp η
dη =
zIpη + b
2 tan δp
1
2 η2
ηO
ηI