andA(x„ , x p ) and B (x„ , x B (are 

 c o K o c o K o 



I /di T /DV /di T /D\ d / \ 



= — + + 2 I l(x - 0.5) {— sin0 p 



4 y<ix R ) y XR / < dx R \D "7 



"d /c V 



— I — sin 0p| 



dx R \d 7 



+ (x -0.5) 2 



X R 3e < x c - x r) X 



r, :<- — sin p j — [-sin0p](x c -O.5) + 



I I — sin <b 



[dx R \d 



COS 0p 



'<»c-»r> 



3x D 



(99) 



The linearized integrand F, for integration over the blade 

 reference surface, then becomes, in the general form (where 

 a:, bj, and ?j, depend on the particular case of loading or 

 thickness): 



F = (x r - x r ) 



3 f l 



i = 1 \ (1 00) 



[(x R -x Ro )a,+(x^-x c )b j J e, 



[a(x r - x Ro ) 2 + B( x R - x Ro ) (x Cj - x c ) + (jgj (x Cq - x c ) 2 j 3/ 



3 

 = J < a i D l + b , D 2> !i 



i = 1 

 where 



*-L 



1 (x c -x c )(x R -x R )dx R 



[p(x R -x Ro .x Co -x c ,] 



[«$-*] 



(101) 



B(l-xp ) + 2 



(b) ( V x ° 



vf 



A ( 1 - x R{ )' + B ( 1 - x Rq ) (x Cq - x c ) + f^j (x Cq - x/ 





B(x h -x Ru 



>«$ 



(x ( 



o-V 











" 





/ A(x h- 



x Ro ) 2 + B(x h - 



x R )(x - 

 K o c o 



X 



'♦(ft 



)" 



(X 



: o 



x c 



» 2 





(x„ - x J' dx c 



Tp(x R - x R o . X c o -" t )l" 



hfeM 



(102) 



2A(l-x D ) + B(x„ -x„) 



JA(l-x R ) 2 + B(l-x R )(x c -x c )+ ^ (x,. -x c ) 2 



2A(x h -x R ) + B(x, -xJ 



I 



\(x h -x Ro ) 2 + B(x h -x Ro )(x Co -x c ) + (^) (x Co -x c ) 2 



At the singular point x -» x 



I 



■^B + 2b, A] e f 



F (x. ; x R , x r ) = 4 



VA 4A(g'-B 



(103) 



This known value at the singular point allows a straightfor- 

 ward analysis procedure to be undertaken using the procedures 

 previously described. 



Some convergence problems near the leading and 

 trailing edges and over much of the surface for narrow blades 

 (maximum c/D = 0.05) have been resolved by computing 

 the linearized form of F (Equation 100) over the entire blade 

 and adding a correction term which is the difference between 

 the actual integrand and this linear approximation. This 

 option has been included in the computer program and is 

 defined as "linear approximation-plus-difference." When 

 conventional integration techniques are used everywhere 

 except at the singular point, where Equation ( 103) is re- 

 quired, the procedure is defined as "direct." 



For the trailing-vortex sheet, a regular integration can 

 be performed since no singular points occur on the sheet. 

 The strength of the vorticity is given by Equation (59) and 

 the induced velocity field is given by 



T'ij(-^\I w, ^"rV^r (i 



J \ K / b = 1 



04) 



W( 



r o. x R- 9 b>= I 



r o 



-*b 





D 



D 







3 



lo 



.-w b 





D 



D 





dr? 



(105) 



11 



