Balance & Trim

A small shift in balance.
A different response to flight.

Wing + tail · side elevationCalculating

Static margin

Net vertical force

Pitching moment

Current angle of attack

Pitching-moment slope

Response to +3° Constrained pitch rig · seconds

Find trim, then disturb the aircraft.

Balance is an equation.

Trim balances force and moment

L_{wing} + L_{tail} = W

M_{CG} = 0

Symbols & units

$L_{wing}, L_{tail}, W$: Forces, N. Tail lift is signed and may be upward or downward.
$M_{CG}$: Pitching moment about the centre of gravity, N m.

Move the CG with elevator fixed: the lift forces keep their values, but their lever arms change. Find trim again to restore both balances. The tail can produce upward or downward force.

Look at the slope.

A negative dCₘ/dα is restoring. The neutral point gives zero slope. Moving the CG aft of it reverses the response. A trimmed aircraft can still be statically unstable.

KNOW THE MODEL

One degree of freedom.

The response rig holds speed and flight-path direction fixed. It includes an illustrative pitch-damping term. It is not a free-flying trajectory, a phugoid simulation or proof of full-aircraft dynamic stability.

Equations, signs & sources ↓

How the wing, tail and trim are calculated

Geometry and coefficients

h = xCG / c, measured aft from the wing leading edge. Wing aerodynamic centre hAC = 0.25; tail aerodynamic centre ht = 3.4. S = 22 m², c = 1.7 m, ρ = 1.225 kg/m³. The neutral point includes the tail; it is not the wing aerodynamic centre.

Wing, tail and pitching moment

C_{Lw} = 5.4 (α - α_{0})

C_{Lt} = 3.8 (0.65 α + i_{t} + 0.6 δ_{e})

C_{L} = C_{Lw} + κ C_{Lt}

C_{m} = - 0.045 + (h - 0.25) C_{Lw} - κ (3.4 - h) C_{Lt}

Symbols & units

$α, α_{0}, i_{t}, δ_{e}$: Angles in radians. α₀ = −2°, iₜ = −1.5°. Positive elevator is trailing edge down.
$h$: CG position xCG/c, measured aft from the wing leading edge; dimensionless, not altitude here.
$κ$: Tail weighting 0.9 Sₜ/S, dimensionless.
$C_{Lw}, C_{Lt}, C_{L}, C_{m}$: Dimensionless coefficients. Positive pitching moment is nose-up.

Angles are in radians; α₀ = −2°, it = −1.5°, κ = 0.9 St/S. Positive elevator means trailing edge down. Positive moment means nose-up. q̄ = ½ρV², L = q̄SCL, M = q̄ScCm.

Trim and response

Trim solves CL = mg/(q̄S) and Cm = 0 simultaneously. An out-of-range solution is reported, never silently clipped. The teaching envelope is −6° ≤ α ≤ 12° and |δe| ≤ 20°; these are model bounds, not a certified stall or control limit.

Constrained pitch response

\dot{Δ α} = q

I_{yy} \dot{q} = \bar{q} S c [C_{mα} Δ α + C_{mq} \frac{q c}{2 V}]

Speed and flight-path direction are constrained; this is not a free-flight trajectory.

Symbols & units

$q, \bar{q}$: Pitch rate q, rad/s; dynamic pressure q̄ = ½ρV², Pa. These are different quantities.
$I_{yy}, S, c$: Pitch inertia 7000 kg m²; wing area 22 m²; reference chord 1.7 m.
$C_{mα}, C_{mq}$: Static pitching-moment derivative per radian and dimensionless pitch-damping derivative; Cmq = −10.

Iyy = 7,000 kg m² and Cmq = −10 are fixed illustrative values. RK4 integrates the constrained response. It stops outside the angle envelope. Downwash is linear; drag, propulsion moments, fuselage effects, nonlinear stall, structural motion and control-free hinge moments are omitted.

Teaching sources: , Week 3–4, Aerodynamic Forces and Moments, slides 24–43; positive-down elevator convention on slide 11. Independent reference: MIT OpenCourseWare · Static stability and wing/tail contributions. All diagrams and example parameters here are original.