where γ is trim quantity and dm is mean draft. Y'β(0), Y'γ(0), ・ ・ ・, N'ββγ(0) represent hydrodynamic derivatives for even keel condition.

　

　

　

　

　

　

 Figure 7 indicates comparison between predicted and measured results for container ship & VLCC. These ships are involved in the model test results data base. There seems good agreement between predicted and measured results roughly but differences arise as drift angle β becomes larger in Figure 7. Especially on VLCC, the accuracy of prediction becomes worse as value of non-dimensional yaw rate γ' becomes larger.

　

 Figure 8 provides turning trajectories with rudder angle δ = ±35°and the time histories of speed drop ratio U/U0, drift angel β, yaw γ and heading angle ψ for ship A & B. Ship A & B are not included in the model test data base. Symbols in the figure indicate model test results and lines show the results of numerical simulation. These two ships have almost same principal particulars and common fore half hull but they have different aft half hull shape, that is ship A has 'V' form sections and ship B has 'U' form sections in aft half hull. Therefore the inherent characteristics of course stability of both ships are quite different.

　

 It is observable in the figure that measured value of the transfer of ship B in starboard turning is larger than that of ship A. It seems that numerical simulations can represent this characteristics well but there are differences between the measured and simulated values of advance of ship A in port turning and ship B in star-board turning. The peak value of simulated yaw rate γ of ship A is larger than that of ship B same as model test results though there exists slight differences in the peak value of ship A.

　

 From the view points of practical utilization at the design stage, Haraguchi [19, 201 had proposed simplified prediction formulae to evaluate performance indices in the Resolution MSC.137(76) Standards for Ship Manoeuvrability directly using parameters such as the hydrodynamic derivatives, frame line parameters and rudder force coefficients. Using the prediction formulae, the ship designer evaluate ship manoeuvring characteristics without the prediction of hydrodynamic forces acting on a ship.

　

 Course keeping & checking ability and initial turning ability are closely related to the loop width in a spiral characteristics curve and the loop width depends on the stability criterion. Therefore prediction formulae for first & second overshoot angles for 10°/10°zigzag ψ1⁽¹⁰⁾ & ψ2⁽¹⁰⁾, first overshoot angle for 20°/20°zigzag ψ1⁽²⁰⁾ and track reach ST for 10°of heading angle are derived from stability criterion,