Ship turning calculator
What this calculates: steady turn radius, advance, transfer, and tactical diameter from a first-order Nomoto yaw model (K, T indices), with presets for a container ship, a tanker and a naval surface combatant fitted to published 35° free-running model tests.
What it assumes: a constant-speed, step rudder deflection driving a linear first-order yaw response, with the ship moving along its heading — the standard textbook Nomoto simplification, not a full manoeuvring (MMG) model.
What it ignores: speed loss in the turn, drift angle and sway (so it reads advance low and transfer high against measured turns), roll during the turn, rudder ramp time, and the nonlinearity that makes K′ depend on rudder angle.
Vessel
KRISO Container Ship benchmark hull, L 230 m, CB 0.651. Calibrated to a free-running model test at 35° rudder.
Nomoto indices (K′, T′)
Higher K′ means a tighter steady turn (R = L / (K′·δ)); higher T′ means a slower yaw response and a longer advance. They convert to dimensional K = K′·V/L and T = T′·L/V. With K′ and T′ fixed, speed changes only how fast the turn happens, not its shape in ship lengths. The presets are effective values for a 35° turn.
Use this turn radius and speed in the stability tool →Turning circle figures
| Quantity | Value | In ship lengths |
|---|---|---|
| K (dimensional gain) | 4.060e-2 /s per rad | |
| T (dimensional time constant) | 43.5 s | |
| Steady yaw rate | 1.42 °/s | |
| Steady turn radius | 311 m | |
| Advance (90° heading change) | 580 m | 2.52 L |
| Transfer (90° heading change) | 398 m | 1.73 L |
| Tactical diameter (180° heading change) | 728 m | 3.17 L |
Measured at 35° rudder for the Container ship (KCS) preset: advance 3.04 L, tactical diameter 3.16 L, transfer 1.36 L (Yasukawa et al. free-running test, in Kim et al. (2021) Table 7). K′ is fitted to the tactical diameter; with no drift angle the model reads advance low and transfer high.
Honest limits
• This is a tier-1 Nomoto first-order model (K, T indices only): no drift angle, sway, roll, or speed loss during the turn
• With no drift angle, heading and course coincide. Real ships point inside their track, so against measured turns this model under-predicts advance (by about 6-18% for the presets) and over-predicts transfer (by roughly 25-30%)
• Preset K′/T′ are effective indices fitted to one published 35° free-running model test per hull form (KCS, KVLCC2, ONR Tumblehome), not sea-trial data for a specific ship. Model-scale turns are typically somewhat tighter than full scale
• The indices are linear and fitted at 35° rudder, so at smaller rudder angles the model overstates the turning diameter: real hulls are relatively more responsive at small angles
• IMO Res. MSC.137(76) criteria (advance ≤ 4.5L, tactical diameter ≤ 5L) apply at 35° rudder to ships of 100 m and over and all chemical and gas carriers, not to warships; they are shown for reference, since compliance is judged from full-scale trials
• Rudder deflection is a step; the finite rudder rate of the model tests is absorbed into the fitted T′
• Speed is held constant through the turn, and the stability-tool link passes the approach speed. Real ships slow markedly in a hard-over turn, so the heel computed there is on the high side
How it works
The Nomoto model treats yaw rate as a first-order response to rudder angle: T·dr/dt + r = K·δ. K and T are estimated from non-dimensional K′ and T′ indices as K = K′·V/L and T = T′·L/V, so the steady radius is L/(K′·δ) whatever the speed. Solving that response in closed form gives the heading over time, which is then integrated to trace the ship's trajectory; advance, transfer, and tactical diameter are read off that trajectory at 90° and 180° of heading change. Each preset's T′ comes from the Nomoto reach approximation (advance ≈ R + T′·L), and K′ is then set so the model reproduces the measured tactical diameter (KCS: Yasukawa et al., reported in Kim et al. 2021; KVLCC2: Yasukawa & Yoshimura 2015; ONR Tumblehome: Sanada et al. 2019). The turn radius and approach speed can be handed off to the stability calculator to see the resulting heel angle in a steady turn.