Str^m-Tejsen and Chislett 



Direction of Motion of 

 Origin of Axis System. 

 Velocity ,U. 



Fig. 1 - Diagrammatic 

 definition of motion, 

 orientation, and force 

 parameters in terms 

 of body-axis coordi- 

 nates 



which must be defined before simulation studies 

 can be carried out. 



The functions describing the hydrodynamic 

 forces and moments have been developed into a 

 useful form for analysis purposes with the aid of 

 a Taylor expansion of the functions. If the Taylor 

 expansion is limited to the first-order terms, the 

 linearized equations are obtained (3,4). The 

 present stage of development, which enables 

 realistic simulations of ship manoeuvres to be 

 made, is based largely on a third -order Taylor 

 expansion of the functions. Introducing the as- 

 sumptions that (a) forces and moments have ap- 

 propriate port and starboard symmetry except 

 for a constant force and moment caused by the 

 propeller, and (b) there are no second- or higher - 

 order acceleration terms, and that crosscoupling 

 between acceleration and velocity parameters is 

 negligible, the validity of which assumptions has 

 been verified, for instance, by the measurements 

 reported in Ref. 5, the third-order Taylor expan- 

 sion reduces to the expressions 



X = X.G + X, + X Au + 4 X... Au^ + -i X Au^ 



1 



2 '"uu~ 



g uuu 



+ \ X^y + I X,,r2 + I X,,b' + -| X,,^v^Au + I X^^^r^Au + | X^^^S^Au 



+ X„,vr + X.,svS + X,srS + X,_,,vrAu + X.,!;.,vSAu + X^j.rSAu 



(3) 



+ Y..V + 4 Y v^ + 4 Y.._ vr^ + 4 Y.^cvS^ + Y.. vAu + 4 Y.. . vAu^ 



g V vv 2 V r r 



2 ^v? 8' 



2 vuu 



1 



+ Yr + -Y r^ + -Y tv^ + - Y ..rh^ + Y rAu + ^Y mu' 

 r g rrr 2 rvv 2 rSS' ' ru 2 ruu 



+ Y3S + i YgggS^ + i Y,^^Sv2 + i Y.^^Sr^ + Y.^SAu + ^ Y^^^iAu^ 



+ Y...vrl 



(4) 



The corresponding expression for N is obtained by replacing Y by N in Eq. (4). 



Equating the hydrodynamic forces and moments based on Eqs, (3) and (4) 

 with mass and inertial responses, Eqs. (1), then the nonlinear mathematical 

 model finally becomes (4) 



320 



