Yamazaki 



The unsteady propeller forces are classified into "surface forces" and "bear- 

 ing forces," and the former contains "impulse forces" as mentioned earlier. In 

 the above expressions, Eqs. (82) and (83) represent the surface forces, Eqs. (85) 

 represent the bearing forces, and Eqs. (83) particularly represent the impulse 

 forces respectively. . 



According to the result of a previous paper (6) the chordwise mean velocity 

 component W*j(^,-S^) flowing on the propeller blade can be assumed to be ap- 

 proximately independent of s, and so we can express 



W*(?',-Sk) ^ /^xo(^')' + <o(^) = W*(^) 



(87) 



Taking into account that the mean surface of the rudder passes through many 

 helical trailing vortices shed from the propeller blades at the same time, the 

 chordwise mean velocity component v|^(77j,s) flowing on the rudder surface can 

 be approximated to be independent of time, so that we can assume 



vLc^i'S) - ^x(^i) 



(88) 



Thus, substituting Eqs. (87) and (88) into Eqs. (74) and (75), the velocity poten- 

 tials 0pt and 0|t can be obtained as 



^pt 



"^Rt 



^B ~1 k'=lL J<P = 



(89) 



477 





^x(Vx) - v*°(77;) 



Bx| R| 



du' 



(p = 



It is convenient in actual calculations to use Eqs. (89) instead of 4>p^ and 0^^ in 

 Eqs. (74) and (75). 



Let us reduce further the preceding equations to convenient forms for per- 

 forming the actual calculations. Since the viscous velocity components [v*jj](sp) 

 and [v*5](SP) are expressed by periodic functions of s, they are expanded in the 

 series 



^ + 



(SP) 



(SP) 



Ix + "ixj^sp^ - 2^ ^xn(^) 



in[<9„(.f )+5(^)v+5 ] 



(90) 



^ + 



,*0 



(SP) 



L ^*n(^) e " 



where [v*ix](SP) ^^^ [v*e](SP) are approximately independent of Ci- The vis- 

 cous velocity components [v*ix](SR)j [^*iy](SR) and [v*^]^sK) are independent of 

 time s. Then, it is found physically that m*(^j,7]j, s), g^(r]^,u,s), and the induced 



44 



