A Vortex Theory for the Maneuvering Ship 



velocity induced by the wake due to £33, f^^, fg^ has no effect on the diving plane. 

 The square of the effective velocity is consequently 



'*^(n)„^^(T).,^B-^('tr;,"ej 



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

 steady motion: 



The plane is at a zero angle /3 when the lift generated is null, the motion being 

 parallel to the x-axis. Consequently, in the quasi-steady motion, the effective 

 angle /?„ is 



^e = /3+ [foi(t') - fo2Ct')^B + f05Ct')^B]- E ^kofok^t') 



k = 



In the real motion, the velocity induced by the wake is 



2 r r'' A 



E ^kB fok(0+) <^kB(t') +1 ^ fokC-') </>UB(t'-T')dT' 



k = *- 



= 71 aj^Bj fokC"^') <^BCt'-T')dT' , 



k=0 



and the effective angle of attack is fi'^ = fi^- A/S^ , with 



A/3. 



r r 



E ^kB fok(t')-J fokC-^') ^B(t'--^')dr' 



Consequently, the set of hydrodynamic forces acting on the diving plane and 

 fin B is: 



T^^bU'^x„ [1 + 2foo + 2fo2^B - 2fo4^] - 



Yr = , 



Zb = - 9 ^^bU'c, [1 + 2foo + 2fo2^B - 2fo,T)g] 



/? + (fo^ - £02^3) + £057,3 - 2] aicB^ok - A/3, 



873 



