Elevator control power and trim

Why it matters

A stable aircraft still has to be flown at many speeds, weights and CG positions, and the elevator is what lets the pilot choose the trimmed angle of attack. How much pitching moment one degree of elevator produces — elevator control power — decides how much deflection each flight condition needs and fixes the forward CG limit, usually in the landing configuration at maximum lift.

Key ideas

Sign convention. Elevator deflection δe is positive trailing edge down. Trailing-edge-down elevator increases tail lift and pitches the nose down, so the elevator control power Cmδe = ∂Cm/∂δe is negative for a tail-aft aircraft. Up-elevator (negative δe) is needed to trim at higher CL (lower speed).

Elevator effectiveness τ. Deflecting the elevator changes the tail lift as if the whole tail's angle of attack had changed by τ·δe, where τ = dα_t/dδe depends mainly on the ratio of elevator chord to tail chord (about 0.4–0.6 for typical 25–40 % chord elevators; read it from a chart). The tail lift increment is a_t·τ·δe.

Control derivatives.

• Pitching moment: Cmδe = −η·V_H·a_t·τ.
• Lift: CLδe = η·(S_t/S)·a_t·τ — small, because the tail area is small, and often neglected.

Trim. The aircraft is trimmed when the total Cm about the CG is zero at the required lift coefficient. With the linear model CL = CLα·α + CLδe·δe and Cm = Cm0 + Cmα·α + Cmδe·δe = 0, the two equations give the trim angle of attack and elevator angle together. Neglecting CLδe makes the algebra simpler: α = CL/CLα and δe = −(Cm0 + Cmα·α)/Cmδe.

Elevator angle to trim versus CL. With CLδe neglected, dδe/dCL = −(dCm/dCL)/Cmδe = K_n/Cmδe. With Cmδe negative and a positive static margin K_n, the slope is negative: more up-elevator is needed as speed falls. The slope shrinks to zero as the CG reaches the neutral point — the basis of the flight-test method for finding the neutral point.

Forward CG limit. Moving the CG forward increases K_n, so more up-elevator is needed at a given CL. The most demanding case is usually trimming at CL_max with flaps down in ground effect (ground effect reduces downwash at the tail and adds a nose-down moment). The forward CG limit is where the maximum up-elevator deflection just trims CL_max. Tail stall (the tail reaching its own stall angle with large negative α_t) can also limit the usable deflection.

Ways to trim. Elevator with a trim tab (zero stick force), an all-moving or adjustable stabiliser (large transports trim with stabiliser incidence i_t and keep the elevator for manoeuvring), or fuel transfer to move the CG (Concorde).

Limits. Linear aerodynamics; τ falls at large deflections because of flow separation, so real control power is less than the linear value near full deflection.

Formulas

Cmδe = −η·V_H·a_t·τ and CLδe = η·(S_t/S)·a_t·τ

• η: tail efficiency; V_H = S_t·l_t/(S·c̄); a_t: tail lift-curve slope (per rad); τ: elevator effectiveness (dimensionless). Units per rad.

CL = CLα·α + CLδe·δe, Cm = Cm0 + Cmα·α + Cmδe·δe

α_trim = (CL·Cmδe + Cm0·CLδe)/Δ, δe_trim = −(Cm0·CLα + Cmα·CL)/Δ, Δ = CLα·Cmδe − Cmα·CLδe

δe_trim ≈ −(Cm0 + Cmα·CL/CLα)/Cmδe (CLδe neglected)

dδe/dCL = K_n/Cmδe (CLδe neglected; K_n = static margin)

Cm = Cm_ac + CL·(h − h_n) + Cmδe·δe

• Cm_ac: moment coefficient about the neutral point (constant). Used for CG-limit problems.

Worked examples

Example 1 (standard). An aircraft has V_H = 0.5, η = 0.9, a_t = 4.0 per rad, τ = 0.45, Cm0 = 0.06, Cmα = −0.6 per rad and CLα = 5.0 per rad. Neglecting CLδe, find Cmδe and the elevator angle to trim at CL = 0.8.

1. Cmδe = −η·V_H·a_t·τ = −0.9 × 0.5 × 4.0 × 0.45 = −0.81 per rad.
2. α = CL/CLα = 0.8/5.0 = 0.16 rad.
3. δe = −(Cm0 + Cmα·α)/Cmδe = −(0.06 − 0.6 × 0.16)/(−0.81) = −(0.06 − 0.096)/(−0.81) = −(−0.036)/(−0.81) = −0.0444 rad.
4. In degrees: −0.0444 × 57.30 = −2.55°.

Answer: Cmδe = −0.81 per rad; δe_trim = −2.55° (trailing edge up).

Example 2 (GATE level). For the same aircraft, S_t/S = 0.2. (a) Including CLδe, find the trim α and δe at CL = 0.8. (b) Using Cm = Cm_ac + CL·(h − h_n) + Cmδe·δe with Cm_ac = 0.06 and h_n = 0.42, find the most forward CG at which the elevator limit of −25° can still trim CL_max = 1.6 (neglect CLδe and ground effect).

(a)

1. CLδe = η·(S_t/S)·a_t·τ = 0.9 × 0.2 × 4.0 × 0.45 = 0.324 per rad.
2. Δ = CLα·Cmδe − Cmα·CLδe = 5.0 × (−0.81) − (−0.6)(0.324) = −4.05 + 0.1944 = −3.8556.
3. α = (CL·Cmδe + Cm0·CLδe)/Δ = (0.8 × (−0.81) + 0.06 × 0.324)/(−3.8556) = (−0.648 + 0.01944)/(−3.8556) = 0.1630 rad = 9.34°.
4. δe = −(Cm0·CLα + Cmα·CL)/Δ = −(0.30 − 0.48)/(−3.8556) = −0.0467 rad = −2.67°.
5. Check: 5.0 × 0.1630 + 0.324 × (−0.0467) = 0.815 − 0.015 = 0.800 ✓.

(b)

1. δe = −25° = −0.4363 rad, so Cmδe·δe = −0.81 × (−0.4363) = +0.3534.
2. Trim: 0 = 0.06 + 1.6·(h − h_n) + 0.3534 → h − h_n = −0.4134/1.6 = −0.2584.
3. h = 0.42 − 0.2584 = 0.1616.

Answer: (a) α = 9.34°, δe = −2.67°; (b) forward CG limit ≈ 0.162c̄. Including CLδe changed δe by only about 0.13°, which is why it is often neglected; including ground effect would move the limit aft.

Common mistakes

• Sign of δe: trailing-edge-down is positive and gives a nose-down moment, so Cmδe < 0.
• Forgetting τ: the elevator does not change the tail angle of attack one-for-one.
• Using the tail area ratio S_t/S in Cmδe instead of V_H (the moment needs the tail arm).
• Thinking up-elevator is needed for higher speed — it is needed for higher CL, i.e. lower speed.
• Ignoring that the forward CG limit is set at CL_max in landing configuration, not at cruise.
• Mixing degrees and radians when Cmδe is per rad.

For GATE AE

Common questions: compute Cmδe or τ from tail data, find the elevator angle to trim from Cm0, Cmα and CL, use the gradient dδe/dCL to infer static margin or neutral point, and find a forward CG limit from a maximum elevator angle. Practise the 2×2 trim solution and the sign of each derivative.

Quick check

1. What is the sign of Cmδe for a conventional aircraft, and why?
2. η = 1, V_H = 0.6, a_t = 4 per rad, τ = 0.5. Find Cmδe.
3. With K_n = 0.1 and Cmδe = −1.0 per rad, what is dδe/dCL?
4. Which flight condition usually sets the forward CG limit?

Answers: 1. Negative — trailing-edge-down elevator raises tail lift and pitches the nose down. 2. −1.2 per rad. 3. −0.1 rad per unit CL (−5.73° per unit CL). 4. Trimming at CL_max with flaps down near the ground.