A Vortex Theory for the Maneuvering Ship 



In par. 6, we consider the forces acting on the planes and fins. We neglect 

 the effect of the history of their own motion. But we take into account the effect 

 of the velocity due to the wake generated by the body itself and show that it acts 

 so as to increase the efficiency of these appendages at the beginning of a 

 maneuver. 



Paragraph 7 is devoted to the other set of forces acting on the body (fric- 

 tion, gravity, inertia, . . .), and par. 8 gives the total expression of the forces 

 when the motion is parallel to the (z,x)-plane. 



22.4. Paragraphs 9-12 are devoted to motions not parallel to the (x,y)- 

 plane. In par. 9, we explain the difficulties we have encountered in this task. 

 They are partly due to the fact that in the most general case, the field of 

 vortices may be different from the sum of those which we deal with when the 

 number of degrees of freedom is smaller. For instance, at a given instant t ' , 

 perhaps the free vortices are generally shed along a line only and not, simul- 

 taneously, along the two lines which are respectively related to the components 

 of the motion parallel to the (z,x) -plane, and to its components parallel to the 

 (x,y)-plane. Nevertheless, after a discussion, we admit that such an addition 

 is possible in some cases of great importance from a practical point of view, 

 when the perturbations are small. Consequently, we obtain final formula simi- 

 lar to those of par. 8. But it is necessary to consider, that in some cases, par- 

 ticularly when the angle of heel is great, or when the body turns with a small 

 radius of gyration, the nuclei found in the integral expressions of the forces 

 and moments depend not only upon the difference t ' - t', but also upon t' (see 

 par. 13). 



22.5. In Section II (pars. 14-16), we examine the case of steady and har- 

 monic forced motions in the (z.x) -plane. Such a study leads to consider the 

 differences between the case of the quasi-steady motion theory and the theory 

 developed in the previous paragraphs. 



In both cases, it is possible to express the lift, the drag and the moment in 

 phase with the motion in terms which are proportional to the square of the re- 

 duced frequency oL/U , and the lift, the drag and the moment outphase with re- 

 spect to the motion, in terms which are proportional to the reduced frequency 

 itself. But, if we use the quasi-steady motion theory, we find that the coeffi- 

 cients before (c^L/U) ^ or ^L/u are constants; on the contrary, if we take into 

 account the delayed circulation, they depend upon the reduced frequency. 



That leads to define "apparent" coefficients. Those related to the outphase 

 forces and moments, have their limits, for oL, U^ 0, equal to the "true" coef- 

 ficients; the other are equal to their true values only for large values of -L'U. 



Consequently, tests carried out in harmonic forced motions give the pos- 

 sibility to decide whether the effects of the wake are of importance, or may be 

 neglected. Experiments showed that some of the apparent coefficients, those 

 which are not mixed with terms of inertia of the body or with term coming from 

 the rotation of the axis attached to the body, have important relative variations. 

 Experiments show also that the effects of the wake on the stern diving planes 

 and fins are very high. 



905 



