What is the criterion for static longitudinal stability?

A nose-up disturbance in angle of attack must produce a nose-down pitching moment about the CG, i.e. Cmα = dCm/dα < 0. In addition Cm0 must be positive so that the aircraft can trim (Cm = 0) at a positive lift coefficient. Static stability describes only the initial tendency; whether the oscillation decays is a dynamic stability question.

Why is a conventional wing alone usually not enough for a stable, trimmable aircraft?

A positively cambered wing has Cm_ac < 0, so its Cm0 is negative. If the CG is ahead of the ac it is stable but trims only at negative lift; if behind, it is unstable. A tail provides both a stabilising slope and a positive Cm0 through its setting angle. Flying wings get around this with reflexed aerofoils or sweep combined with washout.

What is the horizontal tail volume ratio and why is it used?

V_H = S_t·l_t/(S·c̄): tail area times tail arm divided by wing area times mean aerodynamic chord. It non-dimensionalises the tail's moment-producing capability so that the tail term in Cm is simply η·V_H·CL_t. Designers size tails from typical V_H values (roughly 0.4–0.8 for most conventional aircraft) and then check stability and control.

How does wing downwash affect the tail's contribution to stability?

The wing deflects the flow downward at the tail by ε, which increases with wing lift: ε = ε0 + (dε/dα)α. When the aircraft pitches up by Δα, the tail sees only (1 − dε/dα)Δα, so its stabilising moment is reduced by that factor. With dε/dα of 0.3–0.5 the tail loses a third to a half of its effectiveness, which must be allowed for when sizing it.

What is tail efficiency η and what values does it take?

η = q_t/q is the ratio of dynamic pressure at the tail to free-stream dynamic pressure. Wing wake and fuselage boundary layer usually reduce it to about 0.8–1.0, while a propeller slipstream can push it above 1. It multiplies the whole tail contribution to Cm, so a T-tail placed out of the wake often has η close to 1.

How does the tail incidence relative to the wing affect trim and stability?

The difference i_t − i_w shifts the tail lift at a given wing angle of attack, so it changes Cm0 but not Cmα in the linear model. Setting the tail at a lower incidence than the wing (negative i_t − i_w) makes the tail carry a download at zero wing lift, which gives a positive Cm0 and allows trim at positive lift. Stability (the slope) is set by CG position, V_H, a_t and downwash.

Static longitudinal stability: wing and tail contributions

Q: Why is a conventional wing alone usually not enough for a stable, trimmable aircraft?

A positively cambered wing has Cm_ac < 0, so its Cm0 is negative. If the CG is ahead of the ac it is stable but trims only at negative lift; if behind, it is unstable. A tail provides both a stabilising slope and a positive Cm0 through its setting angle. Flying wings get around this with reflexed aerofoils or sweep combined with washout.

Q: What is the horizontal tail volume ratio and why is it used?

V_H = S_t·l_t/(S·c̄): tail area times tail arm divided by wing area times mean aerodynamic chord. It non-dimensionalises the tail's moment-producing capability so that the tail term in Cm is simply η·V_H·CL_t. Designers size tails from typical V_H values (roughly 0.4–0.8 for most conventional aircraft) and then check stability and control.

Q: How does wing downwash affect the tail's contribution to stability?

The wing deflects the flow downward at the tail by ε, which increases with wing lift: ε = ε0 + (dε/dα)α. When the aircraft pitches up by Δα, the tail sees only (1 − dε/dα)Δα, so its stabilising moment is reduced by that factor. With dε/dα of 0.3–0.5 the tail loses a third to a half of its effectiveness, which must be allowed for when sizing it.

Q: What is tail efficiency η and what values does it take?

η = q_t/q is the ratio of dynamic pressure at the tail to free-stream dynamic pressure. Wing wake and fuselage boundary layer usually reduce it to about 0.8–1.0, while a propeller slipstream can push it above 1. It multiplies the whole tail contribution to Cm, so a T-tail placed out of the wake often has η close to 1.

Q: How does the tail incidence relative to the wing affect trim and stability?

The difference i_t − i_w shifts the tail lift at a given wing angle of attack, so it changes Cm0 but not Cmα in the linear model. Setting the tail at a lower incidence than the wing (negative i_t − i_w) makes the tail carry a download at zero wing lift, which gives a positive Cm0 and allows trim at positive lift. Stability (the slope) is set by CG position, V_H, a_t and downwash.

Static longitudinal stability: the Cmα < 0 and Cm0 > 0 criteria, wing and horizontal-tail contributions, downwash, tail efficiency and tail volume ratio, and finding trim.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Static longitudinal stability decides whether an aircraft disturbed in pitch tends to return to its trimmed angle of attack or diverge from it. The wing and the horizontal tail are the two largest contributors, and their balance fixes where the centre of gravity (CG) may be placed. Every loading chart, tail-sizing calculation and CG-limit check starts from the equations in this topic.

Key ideas

Static stability criterion. An aircraft is statically stable in pitch if a nose-up disturbance in angle of attack produces a nose-down (restoring) pitching moment about the CG. With Cm the pitching-moment coefficient about the CG and α the angle of attack, the requirement is Cmα = dCm/dα < 0. To be able to trim at positive lift with a negative slope, the moment at zero lift must be positive: Cm0 > 0. Static stability is only the initial tendency; whether the motion actually dies out is a dynamic question (short-period and phugoid modes).

Notation. Positions along the mean aerodynamic chord c̄ are written as fractions of c̄ from its leading edge: h for the CG, h_ac for the wing–body aerodynamic centre (about 0.25 at low speed). Nose-up moment is positive.

Wing contribution. The wing lift L_w acts at its aerodynamic centre, where the moment coefficient Cm_ac is constant with α (negative for a cambered aerofoil). Taking moments about the CG gives Cm_w = Cm_ac + CL_w (h − h_ac). With CL_w = a_w α, the wing slope is a_w (h − h_ac). If the CG is behind the aerodynamic centre (h > h_ac) the wing is destabilising; if ahead it is stabilising. A conventional cambered wing alone has Cm0 = Cm_ac < 0 and cannot be trimmed stably at positive lift — this is why a tail (or reflex camber / sweep with twist on flying wings) is needed.

Tail contribution. The horizontal tail of area S_t sits a distance l_t behind the CG. Its lift coefficient CL_t, based on S_t, produces a nose-down moment for positive tail lift. Two effects reduce what the tail sees:

Tail efficiency η = q_t/q, the ratio of dynamic pressure at the tail to free stream (about 0.8–1.0), because of wing wake and fuselage boundary layer, or above 1 in a propeller slipstream.
Downwash ε = ε0 + (dε/dα)·α: the wing deflects the flow downward at the tail, so the tail angle of attack grows more slowly than α. Typical dε/dα is 0.3–0.5. For an elliptic-like wing, ε ≈ 2CL_w/(π·A) (radians), so dε/dα ≈ 2a_w/(π·A) with a_w per radian.

The tail angle of attack is α_t = α_w − ε + (i_t − i_w), where i_w and i_t are wing and tail incidence angles relative to the fuselage reference line. Its stabilising effect is weighted by the horizontal tail volume ratio V_H = S_t·l_t/(S·c̄), typically 0.4–0.8 for transport and general-aviation aircraft.

Combining. The tail moment slope is −η·V_H·a_t·(1 − dε/dα). The tail is always stabilising in this term; it must overcome a destabilising wing (CG aft of ac) plus the fuselage and nacelles (next topic). The tail setting (i_t − i_w) is the main lever on Cm0: setting the tail at a smaller incidence than the wing (negative i_t − i_w) makes Cm0 positive.

Limits. The analysis is linear (attached flow, small α), quasi-steady, rigid aircraft, stick-fixed (elevator held). Tail stall, compressibility, aeroelastic effects and power effects change the numbers.

Formulas

Cm_cg = Cm_ac + CL_w·(h − h_ac) − η·V_H·CL_t

Cm_cg: pitching-moment coefficient about the CG (based on S and c̄); Cm_ac: wing moment coefficient about its ac; h, h_ac: CG and ac positions as fractions of c̄; CL_w, CL_t: wing and tail lift coefficients (each on its own area); η: tail efficiency; V_H: tail volume ratio (all dimensionless).

V_H = S_t·l_t / (S·c̄)

S_t, S: tail and wing areas (m²); l_t: CG (strictly) or wing ac to tail ac distance (m); c̄: mean aerodynamic chord (m).

CL_t = a_t·[α_w·(1 − dε/dα) − ε0 + i_t − i_w]

a_t: tail lift-curve slope (per rad or per degree, matching α); ε0: downwash at α_w = 0.

Cmα = a_w·(h − h_ac) − η·V_H·a_t·(1 − dε/dα)

a_w: wing lift-curve slope. Stable when Cmα < 0. Valid for linear aerodynamics, stick fixed.

Cm0 = Cm_ac + η·V_H·a_t·(ε0 + i_w − i_t) (α measured from the wing zero-lift line)

α_trim = −Cm0 / Cmα

Worked examples

Example 1 (standard). A wing has a_w = 0.08 per degree, Cm_ac = −0.05, h_ac = 0.25. The CG is at h = 0.30. Find the wing contribution to Cmα and comment.

Cmα,w = a_w·(h − h_ac)
= 0.08 × (0.30 − 0.25) = 0.08 × 0.05 = +0.0040 per degree.

Answer: Cmα,w = +0.0040 per degree — positive, so the wing alone is destabilising with the CG 5 % c̄ behind its ac. Its Cm0 is −0.05, so the wing alone also cannot trim at positive lift.

Example 2 (GATE level). Add a horizontal tail to the wing of Example 1 with S_t = 6 m², l_t = 5 m, S = 30 m², c̄ = 2 m, a_t = 0.07 per degree, η = 0.9, dε/dα = 0.4, ε0 = 0 and i_t − i_w = −3°. Find V_H, Cmα, Cm0 and the trim angle of attack.

V_H = S_t·l_t/(S·c̄) = (6 × 5)/(30 × 2) = 30/60 = 0.50.
Tail slope: −η·V_H·a_t·(1 − dε/dα) = −0.9 × 0.50 × 0.07 × (1 − 0.4) = −0.0189 per degree.
Cmα = 0.0040 − 0.0189 = −0.0149 per degree (stable).
Cm0 = Cm_ac + η·V_H·a_t·(ε0 + i_w − i_t) = −0.05 + 0.9 × 0.50 × 0.07 × (0 + 3) = −0.05 + 0.0945 = +0.0445.
α_trim = −Cm0/Cmα = 0.0445/0.0149 = 2.99°, giving wing CL ≈ 0.08 × 2.99 = 0.239.

Answer: V_H = 0.50, Cmα = −0.0149 per degree, Cm0 = +0.0445, α_trim ≈ 2.99°. Both stability conditions are satisfied.

Common mistakes

Forgetting the factor (1 − dε/dα): downwash weakens the tail's stabilising effect, often by 30–50 %.
Mixing per-degree and per-radian lift slopes in one equation.
Using the tail lift coefficient on wing area: CL_t is based on S_t, and V_H converts it.
Thinking a positive Cm0 alone guarantees stability. You need Cmα < 0 as well; Cm0 > 0 only makes trim at positive lift possible.
Sign of the CG term: CG aft of the ac (h > h_ac) makes the wing destabilising, not stabilising.
Treating η and dε/dα as exact: they come from data sheets or wind-tunnel tests; take them as given data.

For GATE AE

Typical questions: compute Cmα from given wing and tail data, find the tail volume ratio, decide stable/unstable from the sign of Cmα, find the trim angle from Cm0 and Cmα, and reason about how tail setting or CG shift changes Cm0 and Cmα. Practise keeping units consistent (per degree versus per radian) and remembering the downwash factor.

Quick check

State the two conditions for a trimmable, statically stable aircraft.
A wing has a_w = 5 per rad and the CG is 0.04c̄ ahead of the ac. What is the wing Cmα?
What does the factor (1 − dε/dα) represent?
Compute V_H for S_t = 4 m², l_t = 6 m, S = 20 m², c̄ = 1.5 m.

Answers: 1. Cmα < 0 and Cm0 > 0. 2. 5 × (−0.04) = −0.20 per rad (stabilising). 3. The reduction of tail angle-of-attack change caused by wing downwash. 4. 24/30 = 0.80.

Interview questions

All Aircraft Stability and Control interview questions

Try answering each one aloud before you open it.

1.What is the criterion for static longitudinal stability?Concept
A nose-up disturbance in angle of attack must produce a nose-down pitching moment about the CG, i.e. Cmα = dCm/dα < 0. In addition Cm0 must be positive so that the aircraft can trim (Cm = 0) at a positive lift coefficient. Static stability describes only the initial tendency; whether the oscillation decays is a dynamic stability question.
2.Why is a conventional wing alone usually not enough for a stable, trimmable aircraft?Concept
A positively cambered wing has Cm_ac < 0, so its Cm0 is negative. If the CG is ahead of the ac it is stable but trims only at negative lift; if behind, it is unstable. A tail provides both a stabilising slope and a positive Cm0 through its setting angle. Flying wings get around this with reflexed aerofoils or sweep combined with washout.
3.What is the horizontal tail volume ratio and why is it used?Concept
V_H = S_t·l_t/(S·c̄): tail area times tail arm divided by wing area times mean aerodynamic chord. It non-dimensionalises the tail's moment-producing capability so that the tail term in Cm is simply η·V_H·CL_t. Designers size tails from typical V_H values (roughly 0.4–0.8 for most conventional aircraft) and then check stability and control.
4.How does wing downwash affect the tail's contribution to stability?Concept
The wing deflects the flow downward at the tail by ε, which increases with wing lift: ε = ε0 + (dε/dα)α. When the aircraft pitches up by Δα, the tail sees only (1 − dε/dα)Δα, so its stabilising moment is reduced by that factor. With dε/dα of 0.3–0.5 the tail loses a third to a half of its effectiveness, which must be allowed for when sizing it.
5.What is tail efficiency η and what values does it take?Concept
η = q_t/q is the ratio of dynamic pressure at the tail to free-stream dynamic pressure. Wing wake and fuselage boundary layer usually reduce it to about 0.8–1.0, while a propeller slipstream can push it above 1. It multiplies the whole tail contribution to Cm, so a T-tail placed out of the wake often has η close to 1.
6.How does the tail incidence relative to the wing affect trim and stability?Concept
The difference i_t − i_w shifts the tail lift at a given wing angle of attack, so it changes Cm0 but not Cmα in the linear model. Setting the tail at a lower incidence than the wing (negative i_t − i_w) makes the tail carry a download at zero wing lift, which gives a positive Cm0 and allows trim at positive lift. Stability (the slope) is set by CG position, V_H, a_t and downwash.

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