Pien and Strom-Tejsen . ,. , 



Blade Loading Function 



For a two-dimensional lifting surface it is advantageous to express the 

 chordwise loading distribution by a Birnbaum series, because there is a one- 

 to-one correspondence between the loading and downwash terms. In a three- 

 dimensional- propeller problem, however, there is no such advantage. On the 

 other hand, since the kernel functions are expressed in terms of d - 4>, it is 

 possible to carry out the chordwise integration over the blade functionally if the 

 chordwise loading variation can also be expressed in terms oi e - 4>. This is 

 done as follows: , r. „ ,, ^^ -^ , ,, .•: ,•, s 



On the propeller blade <f is a function of p and e, and the pressure jump 

 Ap. representing the blade loading is a function of p and only. Since a given 

 function can be approximated by a polynomial, we write 



^ / ^ - ^lV 



'I Ap.(p,0) . 2] ^n(^)(^j7T^l • \.. (64) 



where e-^ and ^l are the angular coordinates of the trailing and leading edge 

 respectively and are functions of p. The distance along the chord from the lead- 

 ing edge normalized by the chord length is used as the chordwise variable. 



Since {9 - e^)"" ^ [6^- 0- (^l ~ <^)] "> Eq. (64) can be expressed as 



' ^^- ' '" ''"''■■•■'''' " Ap.(p,e) = ^ b^(p)(5-0)" . .. ■;•;"■ vj/ (65) 



n = 



To include the unsteady case we write a general loading function as 



Ap.(p,6?) = Ap.(/3,0)3 + iApi(p,£?)^,, (66) 



where Ap. (p,^)^ and Ap. (p,^)^^ are the real and imaginary components of the 

 loading amplitude distribution. 



Replacing L {^,p,e) in Eq. (51a) by Ap.(p,0) of Eq. (66) we obtain 



u(x,r,0) = J"J[Ap.(p,e)^ + iAp.(p.0)b] [K^^+iKib] dpd0 



||[APi(P,e),Ki^ - Ap.(p,5)bKib] dpdO 

 s 



||[Ap.(p,0)^Kij, + APi(p,e)bKiJ dpdB 



+ 1 



108 



