Aircraft equations of motion and small-disturbance theory

Six-degree-of-freedom rigid-body equations in body axes, Euler-angle kinematics, and small-disturbance linearisation into decoupled longitudinal and lateral-directional sets.

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Why it matters

Static stability tells you only the initial tendency. To know whether a disturbance dies out, how fast, and how the aircraft responds to controls, you need the equations of motion. The full equations are non-linear and coupled; small-disturbance theory linearises them about a steady flight condition, splits them into longitudinal and lateral sets, and turns flight dynamics into eigenvalue problems — the basis of mode analysis, flying-qualities checks and autopilot design.

Key ideas

Assumptions. Rigid aircraft with constant mass; flat, non-rotating Earth as an inertial frame; aircraft symmetric about its x–z plane (so Ixy = Iyz = 0, only Ixz ≠ 0); body axes with origin at the CG; spinning rotors neglected.

Six degrees of freedom. Three translations (body velocities u, v, w) and three rotations (rates p, q, r). Forces X, Y, Z and moments L, M, N are aerodynamic plus thrust; gravity is added explicitly.

Why the extra terms appear. Body axes rotate with the aircraft, so the acceleration seen in them includes Coriolis-type terms such as qw − rv. Similarly the angular-momentum equations contain gyroscopic cross terms like qr(Iz − Iy) and the product of inertia Ixz couples roll and yaw.

Attitude: Euler angles. The orientation relative to Earth axes is given by yaw ψ, then pitch θ, then roll φ (in that order). Euler-angle rates are NOT the body rates; they are related by the kinematic equations below. They become singular at θ = ±90°, which is why simulations of aerobatic aircraft use quaternions.

Small-disturbance theory. Each variable is written as a reference (trim) value plus a small perturbation: u = u₀ + Δu, q = 0 + Δq, θ = θ₀ + Δθ, etc. The reference is steady, symmetric, wings-level flight (v₀ = p₀ = q₀ = r₀ = φ₀ = 0); in stability axes also w₀ = 0. Products of perturbations are dropped, sin Δθ ≈ Δθ, cos Δθ ≈ 1. The aerodynamic increments are expanded as a Taylor series in the perturbations — this is where stability derivatives (Xu, Zw, Mq, …) enter (next topic).

Decoupling. Because the aircraft is symmetric and the reference flight is symmetric, the linear equations split into two independent sets:

  • Longitudinal (symmetric): Δu, Δw (or Δα), Δq, Δθ — controlled by elevator and throttle. Modes: short period and phugoid.
  • Lateral-directional (asymmetric): Δv (or Δβ), Δp, Δr, Δφ — controlled by aileron and rudder. Modes: roll subsidence, spiral and Dutch roll. Each set is a 4th-order linear system ẋ = A·x + B·u. The eigenvalues of A give the modes; negative real parts mean dynamic stability.

Validity. Good for perturbations of a few degrees and a few percent of speed. Not valid for stalls, spins, large-amplitude manoeuvres or when the reference is itself unsteady; non-linear simulation is needed there.

Formulas

Force equations (body axes): X − m·g·sin θ = m·(u̇ + q·w − r·v) Y + m·g·cos θ·sin φ = m·(v̇ + r·u − p·w) Z + m·g·cos θ·cos φ = m·(ẇ + p·v − q·u)

  • X, Y, Z: aerodynamic + thrust forces (N); m: mass (kg); u, v, w (m/s); p, q, r (rad/s).

Moment equations (symmetric aircraft): L = Ix·ṗ − Ixz·ṙ + q·r·(Iz − Iy) − Ixz·p·q M = Iy·q̇ + r·p·(Ix − Iz) + Ixz·(p² − r²) N = −Ixz·ṗ + Iz·ṙ + p·q·(Iy − Ix) + Ixz·q·r

  • Ix, Iy, Iz, Ixz: moments and product of inertia (kg·m²).

Euler-angle kinematics: φ̇ = p + (q·sin φ + r·cos φ)·tan θ, θ̇ = q·cos φ − r·sin φ, ψ̇ = (q·sin φ + r·cos φ)/cos θ

Linearised longitudinal set (stability axes, w₀ = 0): ΔX − m·g·cos θ₀·Δθ = m·Δu̇ ΔZ − m·g·sin θ₀·Δθ = m·(Δẇ − u₀·Δq) ΔM = Iy·Δq̇, Δθ̇ = Δq

Linearised lateral set: ΔY + m·g·cos θ₀·Δφ = m·(Δv̇ + u₀·Δr) ΔL = Ix·Δṗ − Ixz·Δṙ, ΔN = −Ixz·Δṗ + Iz·Δṙ, Δφ̇ = Δp + tan θ₀·Δr

Worked examples

Example 1 (standard). An aircraft has body rates p = 0.2 rad/s, q = 0.1 rad/s, r = 0.05 rad/s at φ = 30° and θ = 10°. Find the Euler-angle rates.

  1. q·sin φ + r·cos φ = 0.1 × 0.5 + 0.05 × 0.8660 = 0.05 + 0.0433 = 0.0933 rad/s.
  2. φ̇ = p + (q·sin φ + r·cos φ)·tan θ = 0.2 + 0.0933 × 0.1763 = 0.2 + 0.0165 = 0.2165 rad/s.
  3. θ̇ = q·cos φ − r·sin φ = 0.1 × 0.8660 − 0.05 × 0.5 = 0.0866 − 0.025 = 0.0616 rad/s.
  4. ψ̇ = (q·sin φ + r·cos φ)/cos θ = 0.0933/0.9848 = 0.0947 rad/s.

Answer: φ̇ = 0.217 rad/s, θ̇ = 0.0616 rad/s, ψ̇ = 0.0947 rad/s. Note θ̇ is well below q because the aircraft is banked.

Example 2 (GATE level). An aircraft of mass 10 000 kg is in steady level flight (θ₀ = 0) at u₀ = 100 m/s. (a) It is disturbed so that Δq = 0.05 rad/s while Δẇ = 0. Using the linearised Z equation, find the perturbation force ΔZ and the corresponding normal load-factor increment. (b) At another instant the pitch attitude has increased by Δθ = 2° with ΔX = 0. Find the forward acceleration Δu̇.

(a)

  1. With θ₀ = 0: ΔZ = m·(Δẇ − u₀·Δq) = 10 000 × (0 − 100 × 0.05) = −50 000 N (negative Z = upward).
  2. Load-factor increment Δn = −ΔZ/(m·g) = u₀·Δq/g = 5/9.81 = 0.51.

(b)

  1. With θ₀ = 0: ΔX − m·g·Δθ = m·Δu̇ → Δu̇ = −g·Δθ.
  2. Δθ = 2° = 0.03491 rad → Δu̇ = −9.81 × 0.03491 = −0.342 m/s².

Answer: ΔZ = −50 kN (Δn ≈ 0.51 g); Δu̇ = −0.342 m/s² — the nose-up attitude tilts weight backward along x and the aircraft starts to slow, the mechanism behind the phugoid.

Common mistakes

  • Treating θ̇ as equal to q (true only with φ = 0) or ψ̇ as equal to r.
  • Dropping the u₀·Δq term in the Z equation — it is first order, not a product of perturbations.
  • Signs of gravity terms: in z-down axes gravity is +mg·cos θ·cos φ in the Z equation.
  • Forgetting Ixz couples roll and yaw; it cannot be dropped for most aircraft.
  • Expecting longitudinal and lateral sets to decouple when the reference flight is a steady turn or sideslip.
  • Applying linear results to large-amplitude manoeuvres.

For GATE AE

Expect conceptual questions on the assumptions of small-disturbance theory, which variables belong to the longitudinal and lateral sets, the meaning of Euler angles, and short numericals using the kinematic relations (θ̇ = q·cos φ − r·sin φ) or a linearised force equation. Know the six equations well enough to identify each term.

Quick check

  1. List the four state variables of the longitudinal set.
  2. q = 0.2 rad/s, r = 0, φ = 60°. What is θ̇?
  3. Why do the longitudinal and lateral equations decouple?
  4. In stability axes, what is w₀?

Answers: 1. Δu, Δw (or Δα), Δq, Δθ. 2. 0.1 rad/s. 3. The aircraft and the reference flight are symmetric, so symmetric perturbations produce no asymmetric forces to first order (and vice versa). 4. Zero.

Interview questions

All Aircraft Stability and Control interview questions

Try answering each one aloud before you open it.

  1. 1.What are the aircraft equations of motion?Concept

    They are Newton's second law for translation and Euler's equations for rotation of a rigid aircraft, written in body axes fixed at the CG: three force equations in u, v, w and three moment equations in p, q, r, plus kinematic equations relating Euler-angle rates to body rates. Because the axes rotate, the force equations contain terms like qw − rv, and the moment equations contain gyroscopic and Ixz cross-coupling terms. Gravity enters through the Euler angles θ and φ.

  2. 2.Explain the small-disturbance theory in the context of aircraft stability.Concept

    Small-disturbance theory assumes that the changes in an aircraft's motion due to disturbances are small enough that they can be linearized. This allows the use of linear differential equations to analyze the stability and control of the aircraft. The theory simplifies the analysis by focusing on small deviations from a steady flight condition, making it easier to predict the aircraft's response to control inputs and external disturbances.

  3. 3.What happens if the small-disturbance assumption is violated?Application

    If the small-disturbance assumption is violated, the linearized equations may no longer accurately predict the aircraft's behavior. This can lead to incorrect stability and control analysis, potentially resulting in unsafe designs. In such cases, nonlinear analysis methods must be used to capture the true dynamics of the aircraft under larger disturbances.

  4. 4.How do the longitudinal and lateral-directional equations differ?Concept

    The longitudinal set describes symmetric motion in the aircraft's plane of symmetry: perturbations in forward speed u, normal velocity w (or α), pitch rate q and pitch angle θ, driven by X, Z and M and controlled by elevator and throttle. The lateral-directional set describes asymmetric motion: sideslip v (or β), roll rate p, yaw rate r and bank angle φ, driven by Y, L and N and controlled by aileron and rudder. For a symmetric aircraft in symmetric reference flight the two sets decouple when linearised.

  5. 5.What role do stability derivatives play in the linearised equations?Concept

    In small-disturbance theory each aerodynamic force and moment perturbation is expanded as a Taylor series in the motion variables, e.g. ΔM = Mu·Δu + Mw·Δw + Mẇ·Δẇ + Mq·Δq + Mδe·Δδe. The partial derivatives are the stability and control derivatives; they become the coefficients of the linear state matrix whose eigenvalues give the modes. Whether a given sign is stabilising depends on the derivative — Mα must be negative and Nβ positive, for example.

  6. 6.Why is the u₀·Δq term kept in the linearised Z-force equation while q·v is dropped?Numerical

    Linearisation drops products of two small perturbations. The full Z equation contains −q·u; with u = u₀ + Δu this becomes −u₀·Δq − Δu·Δq, and u₀·Δq is first order because u₀ is the large reference speed, so it stays. It represents the centripetal acceleration of a curving flight path. The term p·v is a product of two perturbations and is neglected.

  7. 7.An aircraft has pitch rate q = 0.2 rad/s and yaw rate r = 0 while banked at 60°. What is the rate of change of pitch attitude θ̇?Numerical

    Euler kinematics give θ̇ = q·cos φ − r·sin φ = 0.2 × cos 60° − 0 = 0.1 rad/s. The body pitch rate is not the rate of change of pitch attitude unless the wings are level. In a steady banked turn much of the body pitch rate shows up as heading change instead.

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