Brard 



22.6, In Section IE, pars. 17-21, we examine some possible further devel- 

 opments. 



The equations of motions nearly parallel to the x-axis involve unknown 

 coefficients and functions. The unknown coefficients are those we find in the 

 steady forced motions, the "added masses" and the terms which come from the 

 rotation of the axis attached to the body. The unknown functions are due to the 

 wake; they are the Fourier transforms of the part of the apparent coefficients 

 which depend upon the reduced frequency in the harmonic forced motions. Con- 

 sequently tests in steady and in harmonic forced motion yield, in principle, all 

 the unknown coefficients and functions which are necessary for writing the equa- 

 tions of such motions, although these motions are not harmonic. 



The equations so obtained are not differential, but integro-differential. 

 Consequently, they are more complicated than the differential equations intro- 

 duced by using the classical static derivatives, that is, the theory of the quasi- 

 steady motions. For the naval architects, this new aspect of the problem is 

 somewhat unpleasant and it would be of interest to check whether the errors 

 from the classical treatment of the problem are great or not. Probably they 

 are not negligible in the transient motions. But, until now, we have had no pos- 

 sibility to compare the two families of solutions. Moreover, some people may 

 consider as negligible differences which are important to the eyes of some 

 others. In any case, we think that the views developed in this paper may explain 

 some interesting particularities of the transient motions, because they call the 

 attention to phenomena which prediction would be impossible according to the 

 classical equations. Even if it is finally found that the differences between the 

 solutions of the classical equations and those of the integro-differential equa- 

 tions are not very high, it is of interest to discern why. From this point of 

 view, we think that harmonic forced motion tests are useful, because the results 

 so obtained lead to understand better how the term coming from the partial 

 derivative may partially cancel those coming from the delayed circulation or 

 inversely. 



Some points are yet to be emphasized. Firstly, tests in steady and har- 

 monic forced motions require much care, because of the possible free surface 

 effect in an ordinary tank. Moreover, it is possible that the planar motion 

 mechanisms are not perfectly adapted for systematic research about such 

 motions. For instance, the range of the possible amplitudes and frequencies is 

 probably too narrow. For a point of importance would be to study the limits of 

 the linear field. 



That means that the planar motion mechanisms, which interest has been 

 many times emphasized, do not enable us to solve all the problems involved in 

 the maneuvering qualities of a submerged body. Tests in a steering tank with 

 a rotating arm are certainly necessary in order to explore motions of great 

 amplitude and gyrations at a very large angle of rudder, as was previously the 

 case for researches about the maneuverability of the surface ships. The planar 

 motion mechanisms give new means; but the latter do not replace the previous 

 facilities. It is even allowed to deem that it is necessary to explore the maneu- 

 vering qualities of submerged bodies by using free models as it is already done 

 in the case of surface ships. 



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