Ship Manoeuvring Lab (3D)

Nomoto's model of a turning ship

A ship's yaw rate r responds to the rudder angle delta much like a first-order system. With the non-dimensional yaw rate r' = r L / U and time t' = t U / L (L: ship length, U: speed) the Nomoto-Norrbin equation is T' dr'/dt' + H(r') = K' delta with the hull characteristic H(r') = a r' + b r'^3. K' is the turning ability (rudder gain), T' the sluggishness and the cubic term b limits the turn rate at large rudder angles. For small rudder angles and b = 0 the response to a step is r'(t') = (K' delta / a)(1 - e^(-a t'/T')). The steady turn solves a r' + b r'^3 = K' delta, giving a steady turning diameter D = 2 U_ss / r_ss.

Turning circle, zig-zag, spiral

Turning circle (full rudder, usually 35 degrees): advance is the distance travelled along the original course when the heading has changed by 90 degrees, transfer the sideways offset at that moment, and tactical diameter the offset when the heading has changed by 180 degrees. The IMO standards (MSC.137(76)) require advance of at most 4.5 L and tactical diameter of at most 5 L. Zig-zag (10/10 or 20/20): rudder is put over by the amplitude, reversed when the heading has changed by the same amount, and so on; the overshoot angle after each reversal measures how well the ship checks its swing. The IMO limits for the first overshoot in the 10/10 test are 10 degrees for L/U below 10 s, 20 degrees for L/U of 30 s or more and 5 + 0.5 L/U in between (second overshoot limit 25, 40 or 17.5 + 0.75 L/U). Spiral test (Dieudonne): rudder swept slowly up and down. A directionally stable ship (a above 0) gives a single-valued r'(delta) curve; with a below 0 the curve folds, the ship turns on its own at zero rudder at r' = sqrt(-a/b) and the sweep shows a hysteresis loop of width (2/3) (-a) r_f / K', where r_f = sqrt(-a/(3b)).

Other parts of the model

The rudder moves at a limited rate (default 2.3 degrees per second, as in the SOLAS 35-to-35 degree test). The speed falls in the turn: the equilibrium speed is reduced in proportion to the turn rate (empirical, with a 25 s lag; typical ships lose 30 to 50 percent at full rudder). The drift velocity at the centre of gravity is taken as v = -0.4 L r, pointing outwards, which gives drift angles of 10 to 20 degrees in a tight turn.

Numerical checks on this page

(1) The linear step response is integrated and compared with the closed-form first-order solution after three time constants. (2) Stable hulls: integration to steady state for four rudder angles against the Newton root of the hull curve; unstable hulls: growth of a tiny disturbance to the analytic value sqrt(-a/b). (3) A circle is fitted by three points to the integrated steady track and its diameter is compared with 2 sqrt(u^2 + v^2)/r.

Honest simplifications

Nomoto is a one-degree-of-freedom model fitted to a ship; the K', T', a, b values here are plausible teaching values for a container ship, tanker and ferry, not the data of a real vessel. There is no wind, current, shallow-water effect, propeller wash, heel or full sway-yaw hydrodynamics, and the speed loss and drift are simple empirical relations. A real ship also needs the manoeuvring trial rather than a number from this page; the IMO limits are shown only as a reading aid.

What to try

• Load Container ship: turning circle and read advance, transfer and tactical diameter against the IMO limits; the dashed line is the prediction, the yellow line is live.
• Compare Tanker: slow to respond with Ferry: agile in the zig-zag: the tanker overshoots much more because its time constant is long.
• Load Directionally unstable hull: graph 3 shows the folded steady-state curve and the hysteresis loop; on the zig-zag the overshoot angles become large.
• Switch to Manual rudder, move the rudder slider and try to hold the heading; slow the rudder rate limit to feel the delay.
• Reduce the speed loss to zero and see how much larger the turning circle becomes when the ship does not slow in the turn.