Maneuvering Dynamics
Turning Circle Maneuver
Nonlinear 3-DOF ship manoeuvring model — Abkowitz / Norrbin formulation
The turning circle is the canonical IMO trial that characterises a ship's manoeuvring response to a step rudder command. The simulator integrates the surge–sway–yaw equations of motion forward in time using a polynomial expansion of the hydrodynamic forces in body-fixed velocities and rudder angle:
3-DOF Equations of Motion
$$ \begin{aligned} (m - X_{\dot u})\,\dot u &= X(u, v, r, \delta) \\ (m - Y_{\dot v})\,\dot v + (m x_G - Y_{\dot r})\,\dot r &= Y(u, v, r, \delta) \\ (m x_G - N_{\dot v})\,\dot v + (I_z - N_{\dot r})\,\dot r &= N(u, v, r, \delta) \end{aligned} $$
Forces $X$, $Y$ and moment $N$ expand as polynomials in $u, v, r$ and rudder $\delta$, with coefficients identified from PMM tests.
Key Output Metrics
Three IMO standard quantities are extracted from the simulated trajectory:
Turning Circle Definitions
$$ \text{Advance · Transfer · Tactical Diameter} $$
Advance — x-distance from rudder application to 90° heading change. Transfer — y-distance at 90° heading. Tactical Ø — y-distance at 180° heading change.
Numerical Integration
The state vector $x = [u, v, r, x_E, y_E, \psi, \delta]$ is advanced with explicit Euler at timestep $h$. Rudder command is held at zero until $t = t_0$, then steps to its commanded value $u_i$ to mark the start of the manoeuvre. The integration runs to $T$, sampling all positions and rates along the way.