Aileron control and roll power

Aileron control power and roll damping by strip theory, steady helix angle pb/2V, roll time constant and bank-angle response, adverse yaw and aileron reversal.

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Why it matters

Roll control decides how fast an aircraft can change bank — to enter a turn, to correct a gust upset on short final, or to roll a fighter onto a target. Aileron size is set by a required bank angle in a given time, and the answer depends on the balance between the rolling moment the ailerons produce and the damping the wing provides as it rolls. At high speed, wing twist can make ailerons lose effectiveness or even reverse.

Key ideas

How ailerons work. Ailerons near the wing tips deflect in opposite directions: one trailing edge down (more lift), the other up (less lift). The lift difference acting on a long arm gives a rolling moment. Rolling-moment coefficient Cl = L/(q·S·b); the aileron control derivative Clδa = ∂Cl/∂δa describes control power. (Sign conventions for δa differ between books; here we take magnitudes and the roll toward the up-going aileron.)

Strip-theory estimate. Each spanwise strip under the aileron gains a lift-coefficient increment a·τ·δa, where τ is the aileron effectiveness (as for the elevator, set by aileron chord ratio, read from a chart). Integrating the moment of these strips over the aileron span from y₁ to y₂ on both wings gives Clδa = (2·a·τ/(S·b))·∫ c·y dy. Outboard ailerons are effective because of the large y, but inboard ailerons are used at high speed on transports to limit wing twist.

Roll damping. When the aircraft rolls at rate p, a strip at y sees an extra angle of attack p·y/V — up on the down-going wing, down on the up-going wing. This produces a rolling moment opposing the roll: Clp = ∂Cl/∂(pb/2V), negative. Strip theory gives Clp = −(a/12)·(1 + 3λ)/(1 + λ), i.e. −a/6 for a rectangular wing.

Steady roll rate. After a short transient the aileron moment is balanced by roll damping: Clδa·δa + Clp·(pb/2V) = 0, so the helix angle pb/(2V) = −Clδa·δa/Clp. The helix angle is a non-dimensional roll performance measure. Since pb/2V is fixed by geometry, the roll rate p grows linearly with V in this simple model.

Roll subsidence. The approach to steady roll is first order with time constant τ_R = −I_x/L_p, where L_p = Clp·q·S·b²/(2V) is the dimensional damping. After a step aileron input p(t) = p_ss·(1 − e^(−t/τ_R)), and the bank angle is φ(t) = p_ss·[t − τ_R·(1 − e^(−t/τ_R))].

Adverse yaw. The down-going aileron raises lift and induced drag on the rising wing, which yaws the nose away from the turn (Cnδa usually negative relative to the roll). Cures: differential ailerons (up-going aileron deflects more), Frise ailerons (the up aileron's nose projects into the flow and adds drag), rudder coordination, and spoilers (which reduce lift and add drag on the down-going wing, giving proverse yaw).

Aileron reversal. The aileron's lift increment also produces a nose-down twisting moment on the wing. As dynamic pressure rises, the wing twists enough to cancel the aileron lift; the speed where Clδa becomes zero is the reversal speed. It sets a design condition for wing torsional stiffness and is why spoilers and inboard ailerons are used at high speed.

Limits. Strip theory overestimates (no tip losses or 3-D effects); real values come from lifting-surface theory or tests.

Formulas

Cl = L/(q·S·b)

  • L: rolling moment (N·m); q = ½ρV² (Pa); S: wing area (m²); b: span (m).

Clδa = (2·a·τ/(S·b))·∫ c(y)·y dy (from y₁ to y₂); rectangular wing: Clδa = a·τ·(y₂² − y₁²)/b²

  • a: section lift-curve slope (per rad); τ: aileron effectiveness; c(y): local chord (m); y₁, y₂: inner and outer aileron stations (m).

Clp = −(a/12)·(1 + 3λ)/(1 + λ) (rectangular: −a/6)

pb/(2V) = −Clδa·δa/Clp

  • p: steady roll rate (rad/s); δa: aileron deflection (rad).

L_p = Clp·q·S·b²/(2V), τ_R = −I_x/L_p

  • I_x: roll moment of inertia (kg·m²); τ_R: roll time constant (s).

φ(t) = p_ss·[t − τ_R·(1 − e^(−t/τ_R))]

Worked examples

Example 1 (standard). A rectangular wing has b = 10 m, c = 1.6 m, a = 5.0 per rad, ailerons from y₁ = 3 m to y₂ = 5 m with τ = 0.4. Find Clδa, Clp and the steady roll rate at V = 50 m/s for δa = 15°.

  1. Clδa = a·τ·(y₂² − y₁²)/b² = 5.0 × 0.4 × (25 − 9)/100 = 0.32 per rad.
  2. Clp = −a/6 = −0.833.
  3. δa = 15° = 0.2618 rad. pb/(2V) = −Clδa·δa/Clp = 0.32 × 0.2618/0.833 = 0.1005.
  4. p = 0.1005 × 2V/b = 0.1005 × 100/10 = 1.005 rad/s = 57.6°/s.

Answer: Clδa = 0.32 per rad, Clp = −0.833, pb/2V = 0.10, p ≈ 57.6°/s.

Example 2 (GATE level). A tapered wing has b = 12 m, root chord 2.0 m, taper ratio λ = 0.5 (S = 18 m²), so c(y) = 2.0(1 − y/12). Ailerons run from y₁ = 3.6 m to the tip, τ = 0.45, a = 5.0 per rad. With δa = 20°, V = 60 m/s, ρ = 1.225 kg/m³ and I_x = 6000 kg·m², find Clδa, Clp, the steady roll rate, the roll time constant and the time to reach 30° of bank from a step input.

  1. ∫ c·y dy from 3.6 to 6 = 2.0 × [y²/2 − y³/36] = 2.0 × [(18 − 6) − (6.48 − 1.296)] = 2.0 × 6.816 = 13.632 m³.
  2. Clδa = 2 × 5.0 × 0.45/(18 × 12) × 13.632 = 0.02083 × 13.632 = 0.284 per rad.
  3. Clp = −(a/12)(1 + 3λ)/(1 + λ) = −(5/12)(2.5/1.5) = −0.694.
  4. δa = 0.3491 rad: pb/2V = 0.284 × 0.3491/0.694 = 0.1428; p_ss = 0.1428 × 120/12 = 1.428 rad/s (81.8°/s).
  5. q = ½ × 1.225 × 60² = 2205 Pa. L_p = −0.694 × 2205 × 18 × 144/120 = −33 075 N·m·s. τ_R = 6000/33 075 = 0.181 s.
  6. Bank 30° = 0.5236 rad: solve 1.428·[t − 0.181(1 − e^(−t/0.181))] = 0.5236 → t ≈ 0.539 s (the estimate φ/p_ss + τ_R = 0.367 + 0.181 = 0.548 s is close).

Answer: Clδa = 0.284 per rad, Clp = −0.694, p_ss ≈ 1.43 rad/s, τ_R ≈ 0.18 s, time to 30° ≈ 0.54 s.

Common mistakes

  • Basing Cl on c̄ — lateral moments use span b.
  • Forgetting that both ailerons contribute (the factor 2 in Clδa).
  • Treating Clp as positive; roll damping always opposes the roll.
  • Confusing roll rate p with helix angle pb/2V; the latter is non-dimensional.
  • Using bank angle = p_ss·t straight away and ignoring the roll lag τ_R.
  • Thinking ailerons always get more effective with speed — aeroelastic twist reduces them and can reverse them.

For GATE AE

Expect rolling-moment numericals (Cl·q·S·b), strip-theory Clδa and Clp for rectangular wings, steady roll rate from pb/2V, the roll time constant, and conceptual questions on adverse yaw, differential/Frise ailerons, spoilers and aileron reversal.

Quick check

  1. On what reference length is the rolling-moment coefficient based?
  2. Clδa = 0.2 per rad, Clp = −0.5, δa = 0.1 rad. Find pb/2V.
  3. What is Clp for a rectangular wing with a = 6 per rad?
  4. Name two ways to reduce adverse yaw.

Answers: 1. Wing span b. 2. 0.04. 3. −1.0. 4. Differential or Frise ailerons, spoilers, coordinated rudder.

Interview questions

All Aircraft Stability and Control interview questions

Try answering each one aloud before you open it.

  1. 1.What is an aileron and what role does it play in aircraft control?Concept

    An aileron is a hinged flight control surface usually forming part of the trailing edge of each wing of a fixed-wing aircraft. Ailerons are used to control the aircraft in roll, which is the rotation of the aircraft around its longitudinal axis. By deflecting the ailerons in opposite directions on each wing, the lift on one wing increases while it decreases on the other, causing the aircraft to roll.

  2. 2.Explain roll power (roll control effectiveness).Concept

    Roll power is the rolling moment the ailerons produce per unit deflection, the derivative Clδa = ∂Cl/∂δa, which depends on aileron span and position, chord ratio (through τ) and wing lift slope. What the pilot sees is the steady roll rate, where aileron moment balances roll damping: pb/2V = −Clδa·δa/Clp. Requirements are usually stated as time to reach a given bank angle, which also involves the roll time constant I_x/|L_p|.

  3. 3.How do differential ailerons reduce adverse yaw?Application

    Adverse yaw arises because the down-going aileron on the rising wing increases its lift and induced drag, yawing the nose away from the intended turn. With differential ailerons the up-going aileron (on the descending wing) deflects through a larger angle than the down-going one, adding profile drag on the descending wing and reducing the drag imbalance. Frise ailerons do the same by letting the nose of the up aileron protrude into the airflow.

  4. 4.Why are ailerons usually placed near the wing tips?Application

    The rolling moment of each strip of aileron is its lift increment times its distance y from the roll axis, so outboard ailerons give the most moment for a given area (Clδa ∝ ∫c·y dy). This leaves the inboard trailing edge free for flaps. On large or fast aircraft inboard high-speed ailerons and spoilers are added because outboard ailerons twist the flexible wing tip and lose effectiveness at high dynamic pressure.

  5. 5.How does airspeed influence aileron effectiveness?Application

    For a rigid wing the aileron rolling moment grows with dynamic pressure, but so does roll damping, so the steady helix angle pb/2V is independent of speed and the roll rate p rises in proportion to V. At low speed roll response is therefore sluggish. At high speed wing torsion caused by the aileron's nose-down pitching moment reduces the effective aileron lift; at the reversal speed the net rolling moment is zero, and beyond it the ailerons act in reverse.

  6. 6.Why might an aircraft use spoilers together with ailerons for roll control?Application

    Raising a spoiler on one wing kills lift and adds drag on that wing, so it rolls the aircraft toward it with favourable (proverse) yaw, the opposite of aileron adverse yaw. Spoilers produce little wing twist, so they stay effective at high speed when outboard ailerons suffer aeroelastic loss or reversal, and they free span for larger flaps. Their drawbacks are lag and non-linear response at small deflections.

  7. 7.What is aileron reversal and how is it avoided?Concept

    Aileron reversal is the loss and eventual reversal of roll control at high dynamic pressure because the aileron's lift also twists the wing nose-down, cancelling the lift increment. It is avoided by making the wing torsionally stiff enough that the reversal speed lies well above the design dive speed, by using inboard ailerons near the stiff wing root at high speed, and by using spoilers, which produce little twist.

  8. 8.What is roll damping and what is the roll subsidence mode?Concept

    When the aircraft rolls at rate p, the down-going wing sees an increased angle of attack and the up-going wing a reduced one, producing a rolling moment that opposes the roll — the derivative Clp, which is negative. After an aileron step the roll rate therefore approaches a steady value exponentially with time constant τ_R = I_x/|L_p|, called the roll subsidence mode. Typical time constants are a fraction of a second to about a second.

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