A General Theory for Marine Propellers 



^(k+l)a 



. r y dy , , 



(ay 2 + 2by + c)' 



(k+ l)a 



t> T 2 2c ^ 



a 5 a : 



a (ay 2 + 2by + c)' 



We also need I3* and 13^ for calculating K^^ and K^^. They are 



1/ 



y dy 



1 



by + c 



^° (ay2 + 2by + c)" 



I y dy 



(k+ l)a 



^°- (ay 2 + 2by + c) 



where 



^.-/ 



,( k + 1 ) a 



dy 



Vay^ + 2by + c 

 -^log (^^ + 7ay2 + 2by + c) 



( k+ l)a 



1° 



- 1 / ay + b 



■arc sin 



y^^ 



( k+ l)a 



for a > 



for a < , 



Ii° =— ^log(ay + b) 

 Va 



(k+ l)a 



(46e) 



(47a) 



(47b) 



(48a) 

 (48b) 



for b2 - ac = . (48c) 



Theoretically speaking, in computing Kj^ of Eq. (34b), k takes values from zero 

 to infinity. However for engineering purposes, only a few turns of the path of Q 

 are necessary. Within the range where Q is near to P, and a value of one- half 

 radian can give very good accuracy. This value can be increased while Q is 

 moving away from the point P. By integration stepwise, Ki^ can quite easily be 

 obtained to the desired accuracy. 



In Eq. (34a), the expression for Ki^, the lower integration limit 9 - (p cor- 

 responds to the difference between the angular coordinates of the control point 

 and the field point on the blade Q. The number of integration steps required to 

 obtain the same degree of accuracy for Ki ^ as for Ki^ depends on the blade area 

 ratio. Unless the area ratio is very large, one step is sufficient to obtain a de- 

 sired accuracy. Kn,^, Kn,^, ^n^, and K^^ are computed similarly. 



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