Theory of Unsteady Propeller Forces 



where the x and y axes are chosen so as to coincide with the axis of rotation of 

 the propeller and with the upward vertical line respectively (Fig. 1). The ship 

 is assumed to be advancing straight along the x axis in the negative direction 

 with a constant velocity v as the result of rotating the propeller around the x 

 axis with a constant angular velocity in the negative direction of 0. Further 

 we define another rectangular coordinate system Oj - x^YjZ^ fixed to the ship 

 satisfying the relations 



- vt 



(2) 



where t is time and the origin Oj is set at the representative point on the pro- 

 peller axis. From Eq. (1) and (2), it follows that 



(3) 



RudderlSj,) 

 ro peller Blade(S p) 



Fig. 1 - The ship with a propeller and 

 rudder and the coordinate systems 



Next let us consider the geometrical presentations of positions and shapes 

 of the ship hull, propeller, and rudder. The hull, propeller, and rudder are as- 

 sumed to be set roughly in order from the front. Generally the ship hull contain- 

 ing the propeller hub and bossing or strut is symmetrical with respect to the 

 XjVj plane, and the rudder is set in the vicinity of the Xjy^ plane. The propeller 

 consists of a set of identical, symmetrically spaced blades attached to the hub, 

 having number of blades N, radius Tq, and hub radius rg. Accordingly, the sur- 

 face Sfj of the ship hull can be expressed by 



(-l)''-lzg(Xj,yi) 



(4) 



where 



yfaC'^i) ^ Vi ^ yd('^i) • ''f ^ ''i - ''A - ^oC'^i-yi) - o • 



1,2 



The mean surface Sp of the kih blade of the propeller can be expressed by using 

 the parameters r and v as follows: 



x.(r,v) , r = r , 



,(r) + 9{T) V + 277(k- 1)/N - Qt , 



(5) 



19 



