Prediction of Steering and Manoeuvring of Ships 



X equation: (m-X.)u = fj(u,v,r,S) 



Y equation: (m-Y.)v + (mxQ-Y.)f = fjCu.v.r.S) (5) 



N equation: (mxg -N.)v+ (I2-N.)r= f3(u,v,r,S), 



where 



f,(u,v,r,8) = X, + X> + X^>2 + X^uu^"' + \y 



+ (X^^ + mXc)r2 + X,,h' + >^vvuv'^" + ^rru^-'Au + X^s^S^Au 



+ (X,v + m)vr + X^jvS + X^.rS + X^^^vrAu + X^j^vbAu + X^j^rSAu 



f2(u,v,r,S) 3 Y* + Y,> + Y,^>2 ^ y^^^vrS 



+ Y,v + Y^^^v^ + Y,,,vr2 , y.^^vS^ + Y^^vAu + Y^^^vAu^ 



+ (Y,-mu)r + Y^^^r^ + Y^^^rv^ + Y^,,rS' + Y^^rAu + Y^^^rAu^ 



+ YgS + YssgS^ + Yg^^Sv^ + Yg^^Sr^ + Y^^SAu + Y^.^SAu^ + Ygg^^S^Au 



f3(u,v,r,^) = N* + N*^Au + N^^^^Au^ + N^.^gvrS 



+ N^v + N^^^v^ + N^rrvr' +N^g5vS2 + N^^vAu + N^^^vAu^ 



+ (N,-mxGu)r + N^^^r^ + N.^^rv^ + N^.^rb^ + N^^rAu + N^^^rAu^ 



+ NgS + NsssS^ + Ng^^Sv^ + N.^^&r^ + N,^5Au + N.^^SAu^ + Nsj^^S^Au 



The factorials, 1/2 and 1/6, have been dropped as they unnecessarily com- 

 plicate the bookkeeping, and the fourth-order terms Y^j,^ and N^^^^^ have been 

 introduced to obtain sufficient flexibility of expression for the influence of speed 

 on rudder action. 



Experience has shown that several of the hydrodynamic coefficients or 

 derivatives contained in the above mathematical model are negligibly small. In 

 particular the coefficients which give the changes of the nondimensional coeffi- 

 cients Y,, Y^, Y^, N,, N^, N,, x^^, x^^, x^^, x^s,and X^g with speed are 

 small, as will be discussed later. 



It is not suggested that the above mathematical model represents the ulti- 

 mate in sophistication. It is felt, however, that further significant elaboration 

 will not be justified before advances are made in theoretical or experimental 

 methods of defining the pertinent hydrodynamic coefficients. The next logical 

 step is perhaps to include freedom in roll, which might be necessary for simu- 

 lation of radical manoeuvres of fast warships, which heel appreciably in high- 

 speed turns. Additional terms would of course also be necessary, if it was 

 desired to simulate more complex situations. 



The solution of the mathematical model using a digital computer is de- 

 scribed in detail in Ref. 4. 



321 



