AFTER THE DISTURBANCE

Aircraft Dynamic Stability
& Natural Modes

Disturb the aircraft. Release the controls. Watch what follows.

LONGITUDINAL MOTIONOriginal wing–tail teaching aircraft

02 / ATTITUDE OBSERVATORY · B

Short-period0.0 s

Body axisVelocityLevel reference

Attitude at actual scale · camera centred on aircraft

0.0 / 12 s

Angle of attack · α—Body relative to velocity

Pitch attitude · θ—Body relative to horizon

Flight-path angle · γ—Velocity relative to horizon

True airspeed—Relative to trim

ANo saved case

Preparing the model

Time window

FAST / ANGLE & PITCH RATE

Short-period

↶

Angle of attack and pitch rate respond quickly. Speed changes are usually small on this timescale.

SLOW / SPEED & HEIGHT

Phugoid

∿

Speed and height exchange through the motion. The nose direction alone does not tell you whether the aircraft is climbing.

03 / THREE ANGLES, ONE INSTANT

Where is the nose pointing?

θ = α + γ. The diagram uses actual angles, even when 3D attitude exaggeration is on. Lift acts perpendicular to velocity; drag acts against it.

04 / TRAJECTORY

Follow the displacement

Reference frame

05 / THE SAME CLOCK, THE SAME EQUATIONS

Read the response

Click a graph to scrub time. All values are disturbances from each case’s own trim.
A · dashed amber B · solid turquoise

Angle of attack · Δα / °

Pitch attitude · Δθ / °

Speed · ΔV / m s⁻¹

Height · Δh / m

Pitch rate · q / ° s⁻¹

Elevator input · Δδe / °

Positive = trailing edge down. The pulse is a smooth, one-second trailing-edge-up input, then the elevator returns to its trim setting.

06 / ENERGY IN MOTION · B

Climb, slow down. Descend, speed up.

━ Kinetic━ Potential┄ Total

Energy disturbances per unit mass

Δ e_{k} = V_{0} Δ V

Δ e_{p} = g Δ h

Thrust and drag exchange energy with the aircraft; the sum need not remain constant.

Symbols & units

$Δ e_{k}, Δ e_{p}$: Energy per unit aircraft mass, J/kg. The kinetic term is first order, not the exact finite-amplitude energy.
$V_{0}$: True airspeed at trim, m/s.
$g, Δ h$: Gravitational acceleration, m/s²; height change, m.

Model context and notation

First-order specific-energy changes: Δe_k = V₀ΔV, Δe_p = gΔh. Their sum need not remain constant: thrust and drag can exchange energy with the aircraft. These are linear perturbation energies, not exact finite-amplitude kinetic energy.

07 / HOLD A, EXPLORE B

Compare the response

Quantity	A · saved	B · live

A stays fixed for this page session. Reloading clears it. Export includes model inputs, trim, eigenvalues and sampled responses.

How the model works · equations, limits & sources

Coupled longitudinal motion

Linear motion about the calculated trim

x = {[Δ V, Δ γ, Δ θ, q]}^{T}

\dot{x} = A x + B Δ δ_{e}

\dot{Δ h} = V_{0} Δ γ

The atmosphere and aerodynamic derivatives stay frozen at the freshly calculated level-flight trim.

Symbols & units

$Δ$: Change from the initial trim state. It does not indicate a different aircraft or an absolute angle.
$θ, γ, α$: Pitch attitude, flight-path angle and angle of attack. Angles are radians internally; α = θ − γ.
$Δ V, Δ h, q$: ΔV: m/s; Δh: m; q: pitch rate, rad/s.
$A, B$: Derivatives of the force/moment equations at trim; coefficients retain the corresponding state and input units.

Model context and notation

The state is [ΔV, Δγ, Δθ, q]. We solve ẋ = Ax + BΔδe, with Δḣ = V₀Δγ. A and B are derivatives of planar force and pitch-moment balances at a freshly calculated level-flight trim.

Isolated modes start along the real part of the appropriate complex eigenvector, normalised to the selected initial pitch disturbance. A pulse generally excites a mixture. The periods and damping come from eigenvalues; no animation frequency is prescribed.

An original teaching aircraft

Mass 1,800 kg · wing 22 m² · chord 1.7 m · pitch inertia 7,000 kg m². Wing and tail lift, drag, pitch-rate effects and a fixed body-axis thrust determine the response. CG edits change moment arms and trim; mass and inertia remain fixed. This is a declared parameter study, not a ballast-transfer simulation.

Tail pitch-rate influence changes both lift and moment derivatives. Its slider is not a direct setting of the modal damping ratio.

Respect the small-disturbance boundary

Playback stops before |ΔV| > 10% V₀, |Δα|, |Δθ| or |Δγ| > 5°, |q| > 15°/s, |Δh| > 150 m, or α outside −6° to 12°. Atmosphere and aerodynamic derivatives are frozen at trim. These are teaching bounds, not a certified envelope.

No stall, gust spectrum, full six-degree-of-freedom motion, autopilot or real-aircraft handling-quality claim. Numerical checks do not establish experimental validation.

Learning sources & model record

Week 6, Equations of Motion, Part 2 (27 February 2026), slide 24: longitudinal modes, period and time to half amplitude. Slides 25–27 introduce the lateral modes explored in the second page of this module.

MIT OCW · Aircraft Longitudinal Dynamics. Original figures and geometry; local Three.js under its existing MIT licence.

Model version ·