Abstract: Quasi-stationary hydrodynamic forces are derived from a PMM test with low frequencies, which lead to good agreement with stationary hydrodynamic forces from a stationary straight-line and circular test in the range of exerted kinematic variables in the PMM test. The quasi-stationary hydrodynamic forces are analyzed and characterized by a component-type mathematical maneuvering model proposed by the authors, which lead to good extrapolations of the quasi-stationary hydrodynamic forces exerted in the PMM tests coincident with forces in a stationary straight-line and circular test for a broad range of kinematic variables. The results show that the combination of the PMM test with low frequencies and a component-type mathematical maneuvering model can take the place of a stationary straight-line and circular test. Formulas of the hydrodynamic characteristic constants that occur in the component-type mathematical maneuvering model, and the mathematical maneuvering simulation model with 4 degrees of freedom cooperate to predict ship maneuvering motions in a broad range of kinematic variables.

　

 A ship operating at slow speed in a harbor, during fishing operation in a fishing ground, or during dynamic positioning in the ocean, must keep her heading, position and speed under external disturbances, which are relatively larger than during her cruising speed. Then the ship at her slow speed may be operated with extremely larger drift angle and turning rate due to external disturbances unexpected than at cruising speed. Therefore the ship may fall into dangerous situations for ship operators. For these situations of slow speed maneuvering, the component type mathematical maneuvering model [1][2] which is derived from a vortex system, is available and related to the hydrodynamic forces acting on a ship's bare-hull with a broad range of drift angle and yaw rate like those that occur during harbor maneuvering.

　

 This paper describes a means of PMM-tests analysis which determines quasi-stationary hydrodynamic forces acting on a ship's hull from harmonic captive maneuvering tests. In the procedure, the quasi-stationary hydrodynamic forces abstracted from the tests are compared directly with the stationary hydrodynamic-forces that occur in stationary straight-line and circular tests. Furthermore, the quasi-stationary hydrodynamic forces that occur in PMM tests are analyzed to determine the stationary hydrodynamic-force characteristic constants that occur in a component type mathematical maneuvering model. Additionally, a set of the stationary hydrodynamic -force characteristic constants for the model analyzed permits the good extrapolation of the quasi-stationary hydrodynamic forces beyond the range of kinematic variables in experiments.

　

 Next, this paper describes prediction formulas [3] for the hydrodynamic-force characteristic constants that occur in the component-type mathematical maneuvering model in order to simulate the ship maneuvering motion with 4 degrees of freedom, together with roll. By way of example, stationary hydrodynamic-forces acting on a bare-hull and ship maneuvering motion of the VLCC "Esso Osaka" are predicted and simulated respectively by means of these formulas.

　

 In order to obtain the stationary hydrodynamic forces acting on a ship hull in captive tests precisely, it is very important to define the stationary hydrodynamic forces in the equations of maneuvering motion under a coordinate system with body axes (see Fig. 1).

　

 The equations of motion described in this paper for the stationary hydrodynamic forces are about a midship point as follows.

　

m・(du/dt-v・r-xG・r²)=-mx・du/dt+XH+XE

m・(dv/dt+u・r+xG・dr/dt)=-my・(dv/dt+x1・dr/dt)+YH+YE

(IZZ+m・x²G)・dr/dt+m・xG・(dv/dt+u・r)

=-(JZZ+my・x²t)・dr/dt-my・xt・dv/dt+NH+NE(1)

　

where

　

 Quasi-stationary hydrodynamic forces and yaw moment derived from harmonic captive tests are expressed by the equations of motion mentioned above as follows.

　

XH=(m+mx)・du/dt-m・(v・r+xG・r²)-XE

YH=(m+my)・dv/dt+(m・xG+my・xt)・dr/dt+m・u・r-YE

NH=(IZZ+m・x²G +JZZ+my・x²t)・dr/dt+(m・xG+my・xI)・dv/dt+m・xG・u・r-NE (2)

　

 The quasi-stationary hydrodynamic forces in the left hand side of Eq. (2) are the sum of the acceleration component, velocity component and external force component, e.g. measuring forces. The acceleration component in the right hand side of Eq. (2) is determined by Fourier analysis on raw-data in a harmonic captive test of pure-sway or pure-yaw motion, while in combined motion, additional acceleration tests, e.g. constant acceleration test, are required to estimate the hydrodynamic-force characteristic constants mx, my, JZZ and xt, that occur in acceleration components.

　

 Therefore, quasi-stationary hydrodynamic forces in the time domain are obtained from harmonic captive tests by adding acceleration and velocity components to the external component algebraically, and will be shown in next section.

　

 The quasi-stationary hydrodynamic forces carrying kinematic variables in the time domain are translated to the stationary hydrodynamic forces prescribed by kinematic variables, e.g. drift angle β in pure-sway motion comparable to oblique towing, yaw rate r' in pure-yaw motion comparable to pure turn towing or β and r' in combined motion, and will be shown in the next section.