Abstract: This paper proposes empirical formulae of hydrodynamic coefficients for ship maneuvering motion equation. PMM tests for modern ship hulls including various ship types and drafts are analyzed to improve accuracy of empirical prediction for ship's maneuverability at the initial design stage. For more precise maneuvering analysis, a simple parameter representing stern hull form is newly introduced and included in the empirical formulae. To validate the proposed empirical formulae of hydrodynamic coefficients, the comparison with sea trials and PMM test results have been made. The comparison confirms that the proposed empirical formulae provide improved maneuvering predictions for modern hull forms at the initial design stage.

　

 The requirement to improve the maneuvering capability of ship has been increased among many international societies for the purpose of improving maritime safety and enhancing marine environmental protection. Recently, IMO (International Maritime Organization) has adopted the standards for ship maneuverability in MSC 76[1]. All these movements increase the need for precise prediction of ship maneuvering capability at the initial design stage.

　

 At the initial design stage, empirical formulae for hydrodynamic coefficients based on the principal dimensions of ship have been widely used to simulate ship maneuver. This empirical prediction method using hydrodynamic coefficients is useful in accounting for the effect of principal dimension and rudder specification changes in maneuvering predictions at the initial design stage. This method also is used for drawing up wheel house poster and maneuvering booklet after sea trials.

　

 In this study, for the development of the empirical formulae, PMM test results have been used in statistical analysis for various ship kinds of tanker, bulk carrier, product carrier, container ship and LNG carriers with stern bulb. Model test results at ballast conditions are also considered to improve accuracy of maneuverability prediction at ballast conditions and to confirm the tendency of hydrodynamic coefficients for draft variations which significantly affect on ship maneuverability. From recent research work, stern hull form is known to be a very important factor for ship maneuverability. In this respect, a simple parameter is newly introduced to represent stern hull form. For the validation of newly proposed empirical formulae of hydrodynamic coefficients, comparisons with sea trials have been made. Comparisons with PMM test results are also carried out for validation of tendency according to ship parameter variations. The comparisons confirm that proposed empirical formulae give improved maneuvering predictions for modern hull forms at the initial design stage.

　

 The mathematical model of the ship maneuvering motion is described based on the coupled motion equations of surge, sway and yaw which enable the horizontal motions to be calculated considering coupling effects. Referring to the ship fixed coordinate system on the symmetry plane of the body shown in Fig. 1, the equations of the ship maneuvering motion can be written in the following forms[2]:

　

　

where

　

　

 Subscripts H, P and R correspond to force and moment acting on the hull, propeller and rudder. Prime denotes non-dimensionalized coefficients as shown in the Equation (2).

　

　

where

L:length between perpendiculars

d:mean draft

U:ship speed

p:water density

　

 Non-dimensionalized forces and moments acting on the hull are defined as follows:

　

X'H = X'vrv'r' + X'vvv'² + X'rrr'²

YH = Y'vv' + Y'rr' + Y'vvvv'³ + Y'rrrr'³ + (Y'vrrr' + Yvvrv')v'r' 　(3)

N'H = N'vv' + N'rr' + N'vvvv'³ + N'rrrr'³ + (N'vrrr' + N'vvrv')v'r'

　

 Propeller thrust is defined as follows:

　

　

KT(J) = C1 + C2J + C3J²　(4)

J = u(1-wp)/(nDp)

wp = wpoexp[-4.0(βp-x'pr')]

　

where

t:thrust deduction factor

n:propeller revolution

Dp:Propeller diameter

wp:effective wake fraction

wpo:effective wake fraction in straight running

βp:drift angle at propeller

xp:x-coordinate of propeller location

　

 The rudder force and moment induced by the rudder execution can be defined as follows[3]:

　

X'R = -(1-tR)FNsinδ

Y'R = -(1 + aH)FNcosδ

N'R = -(xR + aHxH)FNcosδ　 (5)

　

　

αR = δ + δO-r(β-l'Rr')

δO = -(πsO/90), sO = 1-u(1-wp)/nP

　

　

K = 0.6/ε

ηH = Dp/H

　

where

　

 ε represents wake ratio between propeller and rudder, and γ denotes rudder straightening coefficient. The coefficient tR is defined as follows[4]:

　

tR = 0.28Cb + 0.55 (6)

　

aH, xH, ε,γ and fα can be evaluated using either the results of empirical formulae or the results of model tests. Added mass and moment can be calculated by the equations proposed by Hooft and Pieffers[5].