SME
Aircraft Design · Problem type
CG in percent MAC and static margin
Use this for “The CG is 4.36 m aft of the datum and the 2 m MAC starts at 3.7 m. Where is the CG in % MAC, and is the airplane stable with hₙ = 0.45?”
Diagram
SM = hₙ − h = 0.45 − 0.33 = 0.12, 12% of c̄ behind the CG
A nose-up gust makes a nose-down moment: the airplane returns.
This is an example. Type your own values to start a new problem.
h
=
xcg (m)
−
xLEMAC (m)
c̄ (m)
SM
=
hₙ
−
h
%MAC
CG in percent MAC
calculated · %
P
Pitch stability
worked out (not typed)
Formulas
h
=
xcg − xLEMAC
c̄
0.33
=
4.36 − 3.7
2
%MAC
=
100
×
h
33 = 100 × 0.33
SM
=
hₙ
−
h
0.12 = 0.45 − 0.33
P
=
stable
if
SM
>
0,
neutral
if
SM
=
0,
else
unstable
0.12 > 0, so stable
Assumptions
•
xcg and xLEMAC are measured from the same datum, + aft.
•
The neutral point hₙ comes from the static-stability analysis of Flight Mechanics.
•
The picture puts the wing’s aerodynamic center at the quarter chord, hac = 0.25.
•
The CG must stay inside its forward and aft limits for every loading.
Step-by-step
WE KNOW
xcg = 4.36 m, xLEMAC = 3.7 m, c̄ = 2 m, hₙ = 0.45
FIND
CG in mean chords (h), CG in percent MAC (%MAC), static margin (SM), pitch stability (P)
Step 1 · Find CG in mean chords (h)
Use h = (xcg − xLEMAC) ÷ c̄
How far the CG sits behind the MAC’s leading edge, in mean chords.
h
=
4.36 − 3.7
2
h
=
0.66
2
h
=
0.33
Step 2 · Find CG in percent MAC (%MAC)
Use %MAC = 100 × h
Write
the
fraction
of
c̄
as
a
percent.
%MAC = 100 × 0.33
%MAC = 33%
Step 3 · Find static margin (SM)
Use SM = hₙ − h
How far the CG sits ahead of the neutral point, in mean chords.
SM = 0.45 − 0.33
SM = 0.12
Step 4 · Find pitch stability (P)
Use P = stable if SM > 0, neutral if SM = 0, else unstable
With the CG ahead of the neutral point, a nose-up gust makes a nose-down moment.
0.12 > 0: stable
P
=
stable
if
0.12
>
0,
neutral
if
0.12
=
0,
else
unstable
P
=
stable
Check
Put the numbers back in. Both sides should match.
0.33
=
4.36 − 3.7
2
✓
33 = 100 × 0.33 ✓
0.12 = 0.45 − 0.33 ✓
0.12 > 0, so stable ✓
Topic check
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