THEORY AND OBSERVATIONS ON THE USE OF 



A MATHEMATICAL MODEL FOR SHIP MANEUVERING 



IN DEEP AND CONFINED WATERS 



Nils H. Norrbin 



Statens Skeppsprovningsanstalt 

 Sweden 



ABSTRACT 



This paper summarizes an experimental and analytical 

 study of ship maneuvering, with special emphasis on 

 the use of a research- purpose slinulator for evaluating 

 the behaviour of large tankers in deep water as well as 

 in harbour entrances and canals. In an introductory 

 Section some new results from full-scale measure- 

 ments and simulator studies are given to illustrate the 

 demands put on a mathematical model in the two ex- 

 treme applications: course-keeping in deep water and 

 manoeuvring in a caunal bend. 



Well-known derivations of rigid body dynamics and 

 homogeneous flow solutions for forces in the ideal case 

 are Included to form skeleton of the mathematical 

 model. Separate equations handle helm and engine 

 controls. Coefficients ajid parameters are made non- 

 dimensional in a new system — here designated the 

 "bis" system as different from the SNAME "prime" 

 system generally used — in which the units for mass, 

 length and time, respectively, are given by the mass 

 of the ship, m, the length, L, and the time required 

 for travelling one s hip length at a speed corresponding 

 to V" = F^L = 1 , VlTg". 



Semi-ennpirlcal methods are suggested for estimates 

 of the force and moment derivatives. Special consider- 

 ation is given to added mass and rudder forces in view 

 of their predominant Importance to course-keeping 

 behaviour; the rudder forces measured on a scale 

 model are corrected for differences In wake and screw 



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