A Vortex Theory for the Maneuvering Ship 



for the values of x inferior to the abscissa of the axis of the gyration here con- 

 sidered (this abscissa is normally positive for the natural gyrations, and, con- 

 sequently, for the forced gyrations which are not too different from the natural 

 ones). 



The component on the z-axis of the velocity induced by the wake is always 

 positive, whatever the sign of v/U + rx/U may be. It seems that it is a quadratic 

 form of the arguments v/U and rx/U, or more exactly of the arguments: 



d / V ' 



dT" iu, 



,( X, t '- t' ) dr' , 



J dr' \u;^, 



g!;,(x, t '- t') dr' , 



when (v U)^, and (Lr/U)^, are continuous for t' >0. Functions (p^(x,t'),4'^(x,t') 

 are null for t ' = 0, and their limits, for t ' = +coj are finite and positive. 



This assumption leads to introduce new functions 4)^(t '), ^^c t ' ) , ^gC t ' ) , 

 sf^C t ' ) null for t ' = , equal to 1 for t ' = +c», and to add to the previous compo- 

 nents of the hydrodynamic forces exerted on the body itself, the components 



t ' * ' / \ 



-ct 



>0 , 



y (5) 



:ALU2 



3 j K)^>;(t'-^')dT'-c; I (^)^^0;(t'-r')dr' 



>0 , 



where c 



3' "-3 ' ^-4' 



' are positive dimensionless coefficients. 



There is also an effect on the diving planes located at the stern. The effec- 

 tive angle of attack /3' = /3 -A/3 (cf. par. 11) becomes 



/3,-A^ -S/3_ with S/3, 



t ' 



T 



(6) 



In this formula, a^^ and a^g are positive dimensionless coefficients, and 



/ 3( t ' ) , .p"^{ t ' ) are null for t ' = and equal to 1 for t ' - +co. 



13. Other Motions of Practical Interest 



Previously we restricted our analysis to the "quasi -rectilinear" motions. 

 But there are other motions of great interest, and particularly, the change of 

 depth and the change of head. 



221-249 O - 66 - 57 



881 



