Manoeuvring Dynamics

Pullout / Yaw-Rate Decay Trial

IMO dynamic-stability test · Nomoto / Norrbin convention

The pullout manoeuvre is the standard diagnostic for course-keeping stability. The ship is steered with a rudder deflection of magnitude $\delta_1$ for a period $T$. Then the rudder is centred ($\delta = 0$). What happens next reveals the ship's intrinsic dynamic behaviour:

Rudder Schedule (both runs)
$$ \delta(t) = \begin{cases} \pm \delta_1, & 0 \leq t < T \\ \, 0, & T \leq t \leq 2T \end{cases} $$

The simulator runs both directions in parallel: $+\delta_1$ and $-\delta_1$, then compares the post-release yaw rates.

Reading the Pullout Signature

The yaw rate $r(t)$ during the rudder-released phase $T \leq t \leq 2T$ is the central diagnostic:

Stability Criterion
$$ \text{Stable} \iff r_{+}(2T) \to 0 \text{ and } r_{-}(2T) \to 0 $$

A dynamically stable hull lets both yaw rates decay to zero after release — the ship straightens out on its own. An unstable hull holds a residual yaw rate, indicating a non-zero steady turn must persist (the hallmark of a course-unstable vessel).

3-DOF Equations of Motion

The underlying solver is the same nonlinear Abkowitz 3-DOF model used by the turning-circle and zigzag pages — surge, sway, yaw equations driven by a polynomial expansion of $X$, $Y$, $N$ in body-fixed velocities and rudder angle. Two independent integrations run with opposite rudder commands.

Simulation Workflow

How the Visualiser Works

Dual trajectory · live rudder · time-compressed playback

+δ Run Port Rudder Path

One ship is integrated with rudder $+\delta_1$ held for $T$ seconds, then released. Its trajectory is drawn in red.

−δ Run Starboard Rudder Path

A second ship is integrated with rudder $-\delta_1$ held for $T$ seconds, then released. Drawn in teal.

3D View Two Ships, Live Rudders

Both ships orbit their tracks simultaneously with their rudders pinned to the actual $\delta(t)$ signal — you can see the rudder snap back to zero at $t = T$.

Replay 5-Second Loop

The full $2T$-second manoeuvre is normalised to a 5-second loop so even slow rudder-decay phases stay visible.

Operating Guide

Simulation Guide

1

Set Ship & Coefficients

Pre-loaded Mariner-class defaults produce a textbook stable pullout. Modify any coefficient to investigate stability margins.

2

Run the Solver

Click Run Simulation. Both directions integrate in parallel for $2T$ seconds and return both yaw-rate histories.

3

Inspect Results

The Pullout Signature plot shows both yaw rates overlaid — read the post-release decay to assess stability. The trajectory plot shows both ship tracks; the Data Table exports the full numeric record.

Manoeuvring · Pullout Stability Trial

Ship Pullout Solver

Operational
Total simulation runs for 2 × T = 1200 s
Hull Mass & Inertia
Added Mass ▾
Linear Damping ▾
Cross-Coupling ▾
Nonlinear Damping ▾
Rudder Coefficients ▾
Bias Terms ▾
Pullout Performance Indices
Final r (+δ run): — °/s
Final r (−δ run): — °/s
Residual Yaw Gap: — °/s
Stability Verdict: —
Samples per run: —
Total Sim Time: — s
READY
U₀: — · δ₁: —
Anim: 0.00 s · Phase: RUDDER APPLIED
Common start
+δ rudder track
−δ rudder track
Rudder released (t = T)