Manoeuvring Dynamics

Dieudonné Spiral Manoeuvre

Steady-state yaw response · IMO course-stability characterisation

The Dieudonné spiral is the most thorough course-stability characterisation in the IMO trials. The ship is run with the rudder fixed at a series of discrete angles (typically $-20°$ through $+20°$ in $5°$ increments) and the steady-state yaw rate is recorded at each setting. The plot of $r_{ss}$ versus $\delta$ — the spiral diagram — is the diagnostic.

Trial Schedule
$$ \delta_j = \{-20°, -15°, -10°, -5°, 0°, +5°, +10°, +15°, +20°\} $$

Nine independent integrations are performed. Each runs for $T$ seconds; the last 200 samples are averaged to give the steady-state yaw rate $r_{ss}(\delta_j)$.

Reading the Spiral Diagram

The shape of the $r_{ss}(\delta)$ curve is the diagnostic for course stability:

Stability Criterion
$$ \text{Stable} \iff \text{single-valued, monotonic } r_{ss}(\delta) \text{ through } \delta = 0 $$

A course-stable hull yields a smooth, monotonic S-curve passing through (or close to) the origin. A course-unstable hull shows a horizontal segment or hysteresis loop near zero rudder — the same small rudder angle can drive multiple distinct steady yaw rates.

Rudder Actuator Lag

The simulator includes a first-order rudder servo model with time constant $T_\delta$:

Rudder Servo Dynamics
$$ T_\delta \, \dot\delta = u_i - \delta $$

This captures the finite response time of real rudder machinery and damps any artificial spike in $\dot\delta$ at $t = 0$.

Simulation Workflow

How the Visualiser Works

Nine trajectories · radial fan replay · time-compressed

9 Cases Parallel Integrations

The backend runs nine independent simulations — one per rudder angle — and returns the full trajectory, heading and yaw-rate histories for each.

Spiral Diagram The Diagnostic

The Plots view defaults to the steady yaw rate $r_{ss}$ vs rudder $\delta$ — the classic spiral curve that reveals stability at a glance.

3D View Radial Fan of Tracks

All nine ship trajectories are drawn from the common origin in a color-graded fan. Nine ship models animate along their respective tracks simultaneously.

Replay 5-Second Loop

The full $T$-second history per case is normalised into a 5-second loop so even slow yaw-rate build-ups stay readable.

Operating Guide

Simulation Guide

1

Set Ship & Coefficients

Pre-loaded with Mariner-class defaults that produce a textbook spiral signature. Adjust mass, inertia and any coefficient to investigate stability margins.

2

Run the Solver

Click Run Simulation. The integrator runs all nine cases sequentially and returns the steady yaw rate spectrum plus full per-case histories.

3

Inspect Results

The Spiral Diagram is the headline plot. Switch trajectories / heading / yaw-rate overlays from the dropdown. The Data Table exports the full numeric record.

Manoeuvring · Dieudonné Spiral Trial

Ship Spiral Solver

Operational
9 cases · last 200 samples averaged for rss
Hull Mass & Inertia
Added Mass ▾
Linear Damping ▾
Cross-Coupling ▾
Nonlinear Damping ▾
Rudder Coefficients ▾
Bias Terms ▾
Spiral Performance Indices
rss @ +20°: — °/s
rss @ −20°: — °/s
rss @ 0°: — °/s
Slope @ origin: — °/s·deg
Stability Verdict: —
Cases / Samples each: — / —
READY
U₀: — · Tδ: —
Anim: 0.00 s · Real t: 0.0 s
Rudder Angle (δ)
−20°0°+20°