SME
Aircraft Design
Weight and balance
Use this for “Empty 3400 kg at 4.5 m, payload 1000 kg at 2 m, fuel 600 kg at 7.5 m. Where is the CG?”
Diagram
M = 3,400 + 1,000 + 600 = 5,000 kg.
The balance point is at x = (3,400 × 4.5 + 1,000 × 2 + 600 × 7.5) ÷ 5,000 = 4.36 m.
On the fulcrum, the turning effects of the masses on each side cancel.
This is an example. Type your own values to start a new problem.
xcg (m)
×
m (kg)
=
mₑ (kg)
×
xₑ (m)
+
mₚ (kg)
×
xₚ (m)
+
mf (kg)
×
xf (m)
Formulas
m
=
mₑ
+
mₚ
+
mf
5,000 = 3,400 + 1,000 + 600
xcg
=
(mₑ
×
xₑ
+
mₚ
×
xₚ
+
mf
×
xf)
÷
m
4.36 = (3,400 × 4.5 + 1,000 × 2 + 600 × 7.5) ÷ 5,000
Assumptions
•
Every arm is measured aft from one datum, often the nose.
•
Each item’s mass acts at its own CG, a point on its arm.
•
Burning fuel moves the CG: rerun with a smaller fuel mass mf.
Step-by-step
WE KNOW
mₑ = 3,400 kg, xₑ = 4.5 m, mₚ = 1,000 kg, xₚ = 2 m, mf = 600 kg, xf = 7.5 m
FIND
total mass (m), CG position (xcg)
Step 1 · Find total mass (m)
Use m = mₑ + mₚ + mf
Add the three masses.
m
=
3,400
+
1,000
+
600
m
=
5,000
kg
Step 2 · Find CG position (xcg)
Use xcg = (mₑ × xₑ + mₚ × xₚ + mf × xf) ÷ m
Each mass times its arm is its moment about the datum; add them, then divide by the total mass.
xcg = (3,400 × 4.5 + 1,000 × 2 + 600 × 7.5) ÷ 5,000
xcg
=
15,300 + 2,000 + 4,500
5,000
xcg
=
21,800
5,000
xcg = 4.36 m
Check
Put the numbers back in. Both sides should match.
5,000 = 3,400 + 1,000 + 600 ✓
4.36 = (3,400 × 4.5 + 1,000 × 2 + 600 × 7.5) ÷ 5,000 ✓
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CG in percent MAC and static margin
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