The Mathematical Model for Merchant Ship Types

A merchant ship has been considered as a rigid body with three degrees of freedom, in the X, Y and Z axes:

That means, translational motion along X and Y and rotation about the Z axis. The ship motions in the other three degrees of freedom have not been included in the mathematical model as they are considered small. The equations of motion in the horizontal plane are given below, References 21:

The left-hand side of the above equations represents the inertia terms and the right-hand side represents the hydrodynamic, rudder, propeller and external forces and moments acting on the ship. The subscript H refers to the hull effect, P refers to the propeller effect, R refers to rudder effects, and E refers to the external disturbances such as wind, wave or current.

The hydrodynamic forces and moments are written in the following form, References 22:

The linear coefficients of the manoeuvring equations are , , , , and . The non-linear coefficients are , , , , , , , , and . X(u) is the ship resistance in ahead motion at forward speed u.

The force developed by the propeller is given by:

where is the propeller thrust and t is the thrust deduction fraction.