Yamazaki 



In calculating the integrals in the equations presented we must take finite 

 parts or principal values for the improper integrals at the singular points or 

 surfaces. 



The angular velocity fi of the propeller is to be determined for a given ve- 

 locity V of the ship so as to satisfy the relation that the difference between the 

 mean thrust of the propeller and the total mean resistance acting on the hull and 

 rudder is equal to zero or to the external force acting on the ship through ropes, 

 etc. On the other hand the nondimensional velocity components v*^, v* , and 

 v*^ are assumed to be given in this paper, though they are to be obtained ex- 

 perimentally or theoretically by using some other procedure. In actual calcula- 

 tions, first substituting Eqs. (25), (26), and (27) into Eqs. (32), (33) and (34), we 

 can obtain the simultaneous equations for m^(fj,T7i,s), gj(^",v,-s^), and 

 gR(77j,u,s) and solve these equations by reference to Eqs. (28) and (29). Then 

 by substituting these solutions into Eqs. (25), (26), and (27) the quantities B0*/3s, 

 w*, Wy, and w* are obtained at an arbitrary point (l,,r],z*) at an arbitrary non- 

 dimensional time s. Accordingly, we can get the pressure p* in the water at 

 (1,^,7], z*) at s from Eq. (36). Further we can calculate the forces and moments 

 acting on the hull, rudder, and propeller by using Eqs. (37) through (56). 



SURFACE FORCES AND BEARING FORCES 



In this section, by applying the theory developed in the previous section, the 

 characteristics of a flow around a ship with a propeller and a rudder will be 

 compared with those around a ship from which the propeller is taken off, and 

 then the mathematical expressions will be derived for the unsteady propeller 

 forces. In the following we shall omit, for the sake of simplicity, the adjective 

 nondimensional for the nondimensional quantities defined in the previous section. 



Let us first consider the case of a ship without a propeller, i.e., a ship 

 composed of a hull and a rudder. Then the quantities in this case are distin- 

 guished from those for the ship with the rudder and the rotating propeller by 

 using the superscript o. Further, we can omit time s, since the quantities are 

 independent of time. For example, m*(^i,77j, s), 0^, Krfxj etc., are expressed 

 as m*°(Ci,nj), c/)^°, k^pj^, etc., for a ship without a propeller. Thus we have 



,* 



0, [v*o] = 0. (57) 



( SR ) 



Hence, from Eqs. (25) and (26), 



where 



k* 



i*o , 0*0 + 0*0 +0*0 _ (58) 



U 2 ^d(^l) . 



•'^F K' = l -^rj^C^j) % (59) 



(Cont) 



32 



