A General Theory for Marine Propellers 



SRjM cos q\ = 3(Xo + XjyXM^ + MjV + M2y^)(ho + hjy + h^y^ + •• • + h y^^) 



where 



Then 



= rp + r^y + r ^y^ + • ■ • + r ^^^^y'^'i'^ , - ' ' •-- , 



Tj = 3(XoMohj + XgMjho + XjMqHo ) , 



r^ = 3(XQMQh2 + XQMjhj+XQM2hg + XjMQhj + Xj+MjhQ) 



etc . 



(61) 



2 q 00 



2 q + 3 ro 



-OS /3q Zi [ h. ^dy + Z r r. ^ dy 



(62) 



By comparing Eq. (58a) with Eq. (32a) it is clear that the evaluation procedure 

 for Kj^ is just the same as that shown in Eq. (45), for Kj, except that there are 

 more terms in Kj^ than in Kj. In the steady case we have I3" and I J*", where n 

 ranges from to 2 and m ranges from to 4. In the case of Kj^ we have n 

 ranging from to 2q + 2 and m ranging from to 2q + 4. 



After Kij, and K^^ have been found, weobtain the expression of K^ by Eqs. 

 (59) and (49a). Likewise we obtain k^ and K^. 



In the evaluation of kj, k^, and K^ we need the following expressions to 

 obtain the additional I3" and Ij": 



,( •<+ 1 )a _ ""__,' ( k+ l)cx 



-I 



y" dy 



(ay^ + 2by + c) 



(2"- 3) b , 



(n- 2) a 3 (n - 2) 



3/2 (n - 2) a r 2 

 (n- 1) 



ay'^ + 2by + c 

 - iq-2 , for n > 2 , 



(63a) 



^1 



( k + 1 )a 



yni dy 



1 



ka / 9 ^, \5/ 2 (m - 4) a 3/ 2 



" (ay2+2by + c) (ay2+2by + c) 



( k + 1 )a 



T in- 1 _ 



(m- 4) a 5 (m - 4) a "5 



for m > 4 . 



(63b) 



We have developed expressions of kernel functions for a pressure dipole 

 that has a strength of e^''^* at a point Q (^,p,d-^t) which moves along a helical 

 line with a constant speed and with its axis normal to the blade element but with 

 no radial component. Furthermore we have shown a step- by- step procedure for 

 evaluating these kernel functions. In each step the evaluation is done functionally. 

 This concludes our discussion of the kernel functions and their evaluations. 



107 



