Theory of Unsteady Propeller Forces 

 A METHOD TO CALCULATE BEARING FORCES 



The mathematical expressions of unsteady propeller forces for a ship with 

 a propeller and a rudder were introduced in the previous section. It is further 

 necessary to reduce them to forms suitable for numerical calculations. How- 

 ever, it seems laborious as shown in the previous section to calculate the sur- 

 face and bearing forces simultaneously by using a high-speed digital computer. 

 Hence, for simplicity, the results presented in the previous section will be re- 

 formulated for bearing forces in this section. Let us denote the components of 

 inflow velocity to the propeller in the x, r, and 6 directions by v*, v*, and v* 

 respectively, which are composed of the viscous velocities and the potential 

 velocities induced by the hull and rudder and are expressed as 



+ V,„ + + V 



* 



^i - B^ 



30*0 ^('^H+l'RC + ^Rt) 

 + V," + 



^ + v*« + —^ ^ ?^ + V* 



' i-de '' ~ ^w '' ^^e '' - (109) 



We shall consider that the propeller is operating in a nonuniform flow with 

 inflow velocity components v*, v*, and v* which are not influenced directly by 

 the presence of the propeller. Further, since the effect of blade thickness on 

 the performance characteristics of the propeller is known to be negligibly 

 small (11), we may neglect this factor in the calculation of bearing forces, i.e., 

 we set 



Pt 



(110) 



Then the following expression for the potential is obtained from Eqs. (74) and 

 (76): 



"Pe 



where 



1 1 N 00 



BX* ' ^'2 30* Jr* ' ' (111) 



®* = <P + ^m(^") + ^(?")v' - ^ + S^' , 



R* - yX*2 + ^'2 + ^'2 _ 2^'^ COS 0* , ' (112) 



55 <^°"" 



