Standard Manoeuvring Trial

20°/20° Zigzag — Course-Keeping & Yaw Response

IMO Resolution A.751(18) · Kempf zigzag · principal yaw-response test

The 20°/20° zigzag (Kempf zigzag) is the IMO-mandated test for evaluating a ship's yaw-checking and course-keeping ability. The rudder is alternated between +20° and −20° at fixed heading thresholds, and the ship's overshoot — how far its heading continues to swing past the threshold after each rudder reversal — quantifies how responsive the rudder is and how much the hull resists changes in heading.

Rudder Switching Logic
$$ \delta(t) = \begin{cases} +20°, & \psi(t) < +20°, \; \dot{\psi} \ge 0 \\ -20°, & \psi(t) \ge +20° \\ +20°, & \psi(t) \le -20° \end{cases} $$

After a steady-state period (length t0) the rudder is first deflected to +20°. From then on the rudder reverses each time the heading crosses ±20°. The ship's heading therefore oscillates as a roughly sinusoidal-but-distorted curve, with peaks beyond ±20° due to inertia.

The Two Overshoots — Acceptance Criteria

Only the first and second overshoot angles determine pass/fail. Both are measured from the trigger threshold, not from zero:

IMO A.751(18) Criteria
$$ \alpha_{1OS} \le \begin{cases} 10°, & L/U_0 < 10 \\ 5° + \tfrac{1}{2}(L/U_0), & 10 \le L/U_0 \le 30 \\ 20°, & L/U_0 > 30 \end{cases} $$

α1OS is the maximum overshoot at the first rudder reversal (when ψ first crosses +20°). The second-overshoot limit is α2OS ≤ α1OS,limit + 15°. Both criteria scale with the time constant L/U0 — large slow ships are permitted more overshoot.

Period & Yaw Rate

The period Tα — time between successive same-direction rudder commands — measures how quickly the ship can be steered. A short period with low overshoot indicates an agile, directionally stable vessel; a long period indicates sluggish response. The peak yaw rate |r|max appears near each rudder reversal and is bounded by the ship's hydrodynamic resistance to rotation.

Simulation Workflow

How the Visualiser Works

Single ship · self-switching rudder · heading-triggered reversals

Phase A Starboard Rudder (+20°)

From t = t0 the rudder is set to +20°. Ship swings to starboard, heading climbs above 0°. Trajectory drawn in amber.

Phase B Port Rudder (−20°)

When ψ ≥ +20° the rudder flips to −20°. The ship continues rotating starboard from inertia (1st overshoot) then swings back. Trajectory drawn in coral.

Threshold ±20° Heading Triggers

The dashed teal lines on the plot mark the heading thresholds. Each crossing flips the rudder. The overshoot above the line is what the IMO criterion measures.

Verdict IMO A.751(18) Tile

The verdict tile evaluates 1st overshoot against the L/U0-scaled limit. ✓ PASS or ✗ FAIL — and the headline metric is the margin in degrees.

Operating Guide

Simulation Guide

1

Set Ship & Conditions

Pre-loaded with Mariner-class defaults. All Abkowitz coefficients are editable in collapsible groups.

2

Run the Solver

Click Run Simulation. The integrator holds rudder amidships until t0, then applies the zigzag schedule for the full simulation time T.

3

Inspect Results

The 3D view animates the S-shaped path. Analytical Plots show heading + rudder overlay, trajectory, yaw rate, and speed loss.

Manoeuvring · IMO Zigzag Trial

Zigzag (20°/20°) Solver

Operational
Time the ship runs straight before rudder is first applied
Initial rudder before t = t₀ — usually 0
Hull Mass & Inertia
Added Mass ▾
Linear Damping ▾
Cross-Coupling ▾
Nonlinear Damping ▾
Rudder Coefficients ▾
Bias Terms ▾
Manoeuvre Constants
Rudder Magnitude 20°
Heading Trigger ±20°
IMO 1OS Limit — °
20°/20° Kempf convention — rudder bound to ±20°, switching at ψ = ±20°.
Yaw Response Performance
1st Overshoot α1OS: — °
2nd Overshoot α2OS: — °
Period Tα: — s
Time to Reach 20°: — s
Max |ψ|: — °
Max |r|: — °/s
IMO A.751(18) Verdict: —
Samples: —
READY
U₀: — · L/U₀: —
Anim: 0.00 s · Rudder: + STBD 20°
Start (rudder amidships)
Rudder + 20° (starboard)
Rudder − 20° (port)
Rudder reversal points
Final position