VERTICAL 



BLADE-REFERENCE 



x AFT 



x„-0.5> 



Fig. 1 Lifting-surface geometry 



The blade surface is given by 



£ 2 =E«,,r) (5) 



= E c (£,,r)±E T (£ 1 ,r) (6) 



where 



E c is the meanline shape, and 



Ej is the thickness shape 



In the analysis, it is convenient to change the variables of 

 integration to(x c , x R ) instead of(£,, r), where 



£, = c(x -0.5) 



= 1 

 r = Dx R /2 



c = chordlength at radius r 

 D = maximum rotor diameter 



(7) 



The position vector of a point on the blade surface 

 described by Equation (5) is 



iw 



(x„ -0.5) sini 



"D COS P I 



e r (0) 



(8) 



and a normal, directed out from the blade surface, is (9, 1 1) 



(9) 



3s 9s 



— =- X — — 

 9x„ 3 x D 



where the plus sign is used for the suction side of the blade 

 and the negative sign for the pressure side of the blade. 

 After some effort it can be shown that 



D 2 f c 3 



N = ± -z-\tt e ? - - 



E/D 



° 



3x„ 



+ N„ e, 



(10) 



c 3 E/D 

 D 3 x R 



di T /D 

 dx 



+ 2(x -0.5) 



d c/D 3 E/D 



d x D 3x, 

 3 E/D 



1° [c 3 E/D 



7l_D cos0 p + ^ sin0 ' > J 



r/c\ 2 E 3E/D1 



[(d) (x c- 05)+ d?7-J 



d P/D cos 0p sin <j>p cos $p 



-£- (x c - 0.5) cos p +JL sin </> p 



X R 



3 E/D 

 3~x~ 



COS 0p 



The normal to the blade reference surface, ^ = 0, < x < 

 K h < X R 



x,, < x R < 1 is 



D 2 c r -| 



N = - + T -5[e 2+ N Ro e r ( eo )j 



di T /D 



N R =2— cos0 p + 2(-)(x -0.5) 



11 o d x R r \D/ L 



(11) 



d P/D cos 2 *p 



d X D 7T Xp 



d Xr 



sin 0p 



N R , the radial component of the normal, is zero for a 

 constant-pitch blade which is neither raked nor skewed. 

 In Equations (10) and (11): 



i T = the total rake 



P = the pitch of the blade 



0p = the pitch angle, <t>p = tan -1 (P/()tx R D)) 



6 = the angular position of a point on the blade 

 surface, a function of both x and x R 



b-1 

 = 2—— jt + S + 2 |— (x r - 0.5) cos ( 



-— sin0 p yx R 



6 % = the skew angle, a function of x R 



= 2 ir + i 



Z 



+ 2— (x c -O.5)cos0 p 



/Xl 



In the derivation of the expressions for numerical analysis, 

 the reference surface (E = 0) is often employed. Generally 

 no specific mention will be made of differences between 

 variables on the blade surface and on the reference surface. 



In a coordinate system rotating with the blades, the 

 fluid velocity may be taken to be the sum of the undisturbed 

 velocity and a component due to the disturbance of the 

 blades: 



