1 T 



Ug(x,r,8) = ^ f J GCp.cJ))^'^ [{r tan S-(x-£) 



k=l 



cos (<|)+6 -6)}-p tan 6 cos (<j)+6 -9)] H d<))dp 



" if J J G s (p ' (J)) ^ T£ 3 [P tan 3 i ° OS ((f>+6 k" 9) 

 r H 9 L k=lB 



(x-£) sin (<H-6 k -8)-r tan $ ] H d<$>dp (18) 



IT Z 



u^(x,r,8) = -|^ I J G(p,<|>)^^ {(x-Q sin ((^-9) 



: H 9 L k=1 



+ p tan 6 sin (tj)+S -6)} H dcfidp 





H -L k=1 



+ (x-O cos (<J)+5 k -6)} H d<f)dp 



where r = hub radius 

 H 



G = - dG(p,(J))/dp = trailing vortex strength in 9 < <j) < °° 

 s ij 



subscripts a, 8, and r indicate axial, tangential, and radial components, 



respectively 



The induced velocity component due to the singularity distributions, normal and 

 tangential to the blade-reference surface, can then be written in a linear approxi- 

 mation from Equations (5) and (7) 



