A Vortex Theory for the Maneuvering Ship 



where 



^B '^ 2 ' "b ^ '^Lj 



2 



k = 



^'-^'b = - 2^'^BLU2(.fgCL -C^ 



fok^t') -| fo,(T') 0,B(t'-r')dr' 



2 r pt' 



E ^kB fok^t')-J fo^(T')c4BCt'-T')dT' 



> (2) 



11.2. Rudders, Ailerons, and Other Appendages 

 Located in the (z,x)-Plane 



Let L^A'O-Oj be the coordinates of the axis o^ of a rudder a; l<^^,o,l^^ 

 those of the center of the lifting surface, which area (aileron included) is a^. 



The absolute velocity of the center of the lifting surface is 



We assume that the components of the incident velocity parallel to the span 

 (hence, to i^) have no effect (this assumption is similar to the hypothesis already 

 made about the diving planes and motivates similar comments). 



According to this assumption, the velocity induced by the wake generated by 

 foo(t'), foi(t') and fgjCt') has no effect on the rudder. The square of the 

 effective velocity is 



1 + 2 



. 2l't?l C, 



The component on the y-axis of the effective incident velocity is, in the quasi- 

 steady motion: 



"" {(u),, " fo4^t')^^A- fo5Ct')CA- a3Afo3(t') - a,Afo4Ct')}- 



The rudder is at a zero angle a when the rudder is in the (z,x) -plane; a is >0 

 when the lift has on the y-axis a negative component. In the quasi-steady 

 motion, the effective angle of attack is 



4 



'^e = '^ + [fo3^t') + fo4^t'),^A- fo5Ct')CA]- E ^KAfokCt'). 



k = 3 



In the real motion, the velocity induced by the wake has on the y-axis a 

 component 



875 



