M42) 



A Vortex Theory for the Maneuvering Ship 



„ ^ „, _ 1 .,,2 2W f Lq , Lw , Lu ^ , L^q ^ 



^i + ^i - 2^'^^ AL {^1 U - '^3-^- Mi3 -^+ '^35 -^ ■' 



Y .Y' 1 An2 2W J Lq , Lw , Lu L^ql 



)lli+?ll. = 2^ALU2- |mi3 + 2A.13 u + (M3 " Mi) - + V33 — + v[^—-K'-^ 



As is well known, the force is null in a steady motion, when Lq/u = 0, but 

 not the moment if the body is not symmetrical with respect to the (x,y)-plane 

 (/Zj3 :|: 0). When q :}= 0, the force and the moment are not null, even when the 

 motion is steady. 



6. Sets of Forces on the Diving Planes and Fins, 

 Effect of the Wake 



The diving planes and fins contribute to the forces f • (par, 5.4) and 5^ 

 (par. 7). We assume here that the expressions of these forces take into ac- 

 count the effect of the diving planes and fins. 



But we have yet to introduce the effects of the lift, moment and drag due to 

 the diving planes and fins. We neglect here the history of their motion because 

 the length of their chord is small with respect to the length of the body itself. 



Let L^g, 0, L^g the coordinates of the axis Og of a diving plane b, L7)g the 

 ordinate of the center of the lifting surface, and erg its area. In -q^ and a^, the 

 fin associated with the plane is assumed to be included. 



When the motion is steady and parallel to the x-axis (w= 0, q = 0), the wake 

 may induce on the plane a velocity which component on the z-axis is w^g = a^^u. 

 The components of the relative incident velocity are 



-U, 0, a„gU. 



Let /3 be the angle of the plane. /3 = when the lift due to the previous inci- 

 dent velocity is null; /? >0, when the lift generated by the plane has a negative 

 component on the z-axis. 



In the most general quasi -steady motion, the relative incident velocity is 



, 



lu)^, Iu/^/b ^oB ^ob (u)^, ^iB (u)j, 



/Lq 

 '2B Vu 



861 



