SME
Aircraft Design · Problem type
Mean aerodynamic chord of a tapered wing
Use this for “A wing spans 15 m with a 2.6 m root chord and a 1.4 m tip chord. Find its mean aerodynamic chord and where it sits.”
Diagram
Drawn
to
scale:
AR
=
b
2
S
=
7.5.
This is an example. Type your own values to start a new problem.
c̄ (m)
=
2
3
×
cᵣ (m)
×
1
+
+
λ
2
1
+
ȳ (m)
=
b (m)
6
×
1
+
2
×
1
+
cₜ
Tip chord
example · m
S
Wing area
calculated · m²
AR
Aspect ratio
calculated
Formulas
λ
=
cₜ
cᵣ
0.5385
=
1.4
2.6
S
=
b
×
cᵣ + cₜ
2
30
=
15
×
2.6 + 1.4
2
AR
=
b
2
S
7.5
=
15
2
30
c̄
=
(
2
3
)
×
cᵣ
×
1 +
λ
+
λ
2
1 +
λ
2.06
=
(
2
3
)
×
2.6
×
1 + 0.5385 +
0.5385
2
1 + 0.5385
ȳ
=
(
b
6
)
×
1 + 2 ×
λ
1 +
λ
3.375
=
(
15
6
)
×
1 + 2 × 0.5385
1 + 0.5385
Assumptions
•
A straight-tapered trapezoidal wing: the chord shrinks evenly from root to tip.
•
The span b runs tip to tip, so ȳ is measured from the root on one half.
•
The wing’s aerodynamic center sits near c̄ ÷ 4 behind the leading edge at ȳ.
Step-by-step
WE KNOW
b = 15 m, cᵣ = 2.6 m, cₜ = 1.4 m
FIND
taper ratio (λ), wing area (S), aspect ratio (AR), mean aerodynamic chord (c̄), station of c̄ (ȳ)
Step 1 · Find taper ratio (λ)
Use λ = cₜ ÷ cᵣ
Taper: the tip chord over the root chord.
λ
=
1.4
2.6
λ
=
0.5385
Step 2 · Find wing area (S)
Use S = b × (cᵣ + cₜ) ÷ 2
A trapezoid: the span times the average of the two chords.
S
=
15
×
2.6 + 1.4
2
S
=
15
×
4
2
S
=
60
2
S
=
30
m²
Step 3 · Find aspect ratio (AR)
Use AR = b² ÷ S
Square the span, then divide by the area.
AR
=
15
2
30
AR
=
225
30
AR = 7.5
Step 4 · Find mean aerodynamic chord (c̄)
Use c̄ = (2 ÷ 3) × cᵣ × (1 + λ + λ²) ÷ (1 + λ)
The
chord
whose
square,
spread
over
the
span,
matches
the
wing’s:
the
integral
of
c
2
over
S
.
c̄
=
(
2
3
)
×
2.6
×
1 + 0.5385 +
0.5385
2
1 + 0.5385
c̄
=
2
3
×
2.6
×
1 + 0.5385 + 0.29
1.5385
c̄
=
2
3
×
2.6
×
1.8285
1.5385
c̄
=
0.6667
×
2.6
×
1.8285
1.5385
c̄
=
1.733
×
1.8285
1.5385
c̄
=
3.169
1.5385
c̄
=
2.06
m
Step 5 · Find station of c̄ (ȳ)
Use ȳ = (b ÷ 6) × (1 + 2 × λ) ÷ (1 + λ)
The
station
on
one
half
where
the
chord
equals
c̄
:
b
6
for
a
triangle,
b
4
untapered.
ȳ
=
(
15
6
)
×
1 + 2 × 0.5385
1 + 0.5385
ȳ
=
15
6
×
1 + 1.077
1.5385
ȳ
=
15
6
×
2.077
1.5385
ȳ
=
2.5
×
2.077
1.5385
ȳ
=
5.1925
1.5385
ȳ
=
3.375
m
Check
Put the numbers back in. Both sides should match.
0.5385
=
1.4
2.6
✓
30
=
15
×
2.6 + 1.4
2
✓
7.5
=
15
2
30
✓
2.06
=
(
2
3
)
×
2.6
×
1 + 0.5385 +
0.5385
2
1 + 0.5385
✓
3.375
=
(
15
6
)
×
1 + 2 × 0.5385
1 + 0.5385
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.