OSCILLATORY / ROLL & YAW
Dutch roll
Sideslip, roll and yaw oscillate together. The yaw damper targets this motion, but it also affects the coupled system.
DYNAMIC STABILITY / CHAPTER 02
Lateral–Directional Stability
& Yaw Damper
One disturbance. Coupled motion. A different response with feedback.
ROLL · YAW · SIDESLIPOriginal small-disturbance teaching aircraft
Dutch roll0.0 s
Nose directionVelocity directionInitial heading
3D is unavailable on this device. The attitude diagrams, trajectory and curves remain available.
Sideslip · β—Velocity relative to nose
Bank · φ—Positive = right wing down
Yaw rate · r—Positive = nose turning right
Rudder · δr—Positive = trailing edge left
Preparing the response
FAST / ROTATION DIES AWAY
Roll subsidence
Wing roll damping removes roll rate quickly. Removing roll rate alone does not guarantee wings-level flight after a general disturbance.
SLOW / BANK DRIFT
Spiral mode
The balance of dihedral and directional effects can make bank decay or grow. A yaw damper is not a bank-hold autopilot.
Mode metrics above describe the airframe with the damper off. Illustrations show characteristic behaviour; the curves below are computed. With feedback on, the same airframe initial state excites the closed-loop system.
03 / FOLLOW THE FEEDBACK · B
A measured rate becomes a rudder motion
MEASUREYaw rate r
→REMOVE SLOW OFFSETWashout · 3 s
→MULTIPLY BY GAINCommand
→ACTUATOR · 0.2 sDamper rudder
A washout filter rejects a sustained yaw rate once its transient has passed. It helps the damper avoid continually opposing a steady turn. This page simulates small disturbances about straight flight; the sketch illustrates the filter’s DC response.
04 / NOSE, VELOCITY & BANK · B
Yaw is not sideslip
Top view: track direction χ ≈ ψ + β. Rear view: positive φ lowers the right wing. Diagrams retain actual angles when 3D exaggeration is enabled.
05 / GROUND TRACK
Follow the lateral displacement
AB
06 / ONE CLOCK, COUPLED EQUATIONS
Read the response
Click a graph or use its arrow keys to scrub time.
A · dashed amber B · solid turquoise
Sideslip · β / °
Bank · φ / °
Roll rate · p / ° s⁻¹
Yaw rate · r / ° s⁻¹
Heading change · ψ / °
Total rudder · δr / °
07 / HOLD A, EXPLORE B
Compare the evidence
| Quantity | A · saved | B · live |
|---|
A remains fixed during this page session. Reloading clears it. Export includes settings, both spectra, inputs and response samples.
How the model works · equations, limits & sources
Coupled lateral dynamics
Coupled small-disturbance lateral motion
The fin, dihedral, damping and control derivatives act together. No longitudinal motion or second-order height loss is calculated.
Symbols & units
- Sideslip and bank, rad; roll and yaw rates, rad/s. Positive roll lowers the right wing; positive yaw turns right.
- Side force, N; rolling and yawing moments, N m. L means rolling moment on this page, not lift.
- Moments of inertia, kg m². Products of inertia are zero in the stated model.
Model context and notation
Aircraft state: [β, p, r, φ]. β̇ = Y/(mV) − r + gφ/V; ṗ = L/Ixx; ṙ = N/Izz; φ̇ = p. The fin’s sideforce acts behind and above the CG. Wing dihedral, roll damping, yaw–roll coupling and aileron adverse yaw complete the model.
First-order heading and ground-track projection
These are projections about straight, level flight, not finite-angle turn geometry.
Symbols & units
- Heading change and sideslip, rad; ψ + β is the first-order track direction.
- Lateral displacement, m; fixed reference true airspeed, m/s.
Model context and notation
Heading ψ̇ = r and lateral displacement ẏ = V(ψ + β) are first-order projections. Speed and altitude are frozen at trim. Products of inertia are set to zero; axes are aligned with the straight, level trim velocity. No second-order height loss is predicted.
A damper, not a bank hold
Washout feedback and finite rudder response
Positive rudder produces a negative yaw moment under this page’s sign convention. The external rudder pulse is added after the damper output.
Symbols & units
- Low-pass yaw-rate state, rad/s; w is not wind or vertical speed here.
- Gain K has units s. Command and damper rudder deflection are rad. Damper OFF sets its command to zero.
Model context and notation
ẇ = (r − w)/3; δcmd = K(r − w); δ̇d = (δcmd − δd)/0.2. K has units of seconds. Positive rudder produces a negative yaw moment, so this feedback sign opposes yaw rate. The external rudder pulse is added to the damper output.
Airframe modes are eigenvectors of the four-state system. The controller adds two states. When switched on, it starts unexcited; it can change other modes as well as Dutch roll. A slowly growing spiral may remain.
Scope and stop boundaries
Original illustrative aircraft: 1,800 kg; wing 22 m²; span 11 m; Ixx 3,000 and Izz 9,000 kg m². This shares teaching dimensions with the longitudinal lab, but is a separately scoped lateral approximation, not a validated combined aircraft.
Stop before |β| > 5°, |φ| > 10°, |ψ| > 15°, |p| or |r| > 15°/s, |δr| or |δcmd| > 10°, or actuator rate > 30°/s. Controls are not silently clipped. No stall, nonlinear spiral dive, turbulence, sustained-turn simulation or handling-quality certification.
Learning sources
Week 6, Equations of Motion, Part 2, 27 February 2026, slides 25–27: lateral modes. Week 13, 17 April 2026, slides 39–52: feedback and flight control.
MIT OCW · Aircraft Lateral Dynamics
MIT OCW · Basic Autopilot Design
Original diagrams and geometry; local Three.js under its existing MIT licence. Numerically checked teaching equations; no flight-test validation. Version .