Lehman and Kaplan 

 N even: m odd or even: Y- (t) = , 



X -ju. . ui :-> ,, X . ( t ) ;^ , i = 1 , . . . , 4 , 



N odd: m even: Y.(t) =0, ' 



.,j. (.11:: , . x.(t) ^ , i = 1, .... 4 , 



m odd: Y. ( t ) / , 



X.(t) = , i = 1, . . . , 4 . 



Thus it is seen that the number of blades, odd or even, determines the nature of 

 the induced forces. In the case of a symmetrically disposed appendage the pre- 

 vious results show that even-bladed propellers induce only axial forces and odd- 

 bladed propellers produce only transverse forces for the fundamental blade rate 

 frequency. Odd-numbered harmonics for an odd number of blades result in only 

 transverse forces, whereas any condition where the product Nm is even results 

 in only axial forces. 



With the vorticity distribution on the propeller given by Eq. (2) the two- 

 dimensional lift on each propeller blade is 



L = 277pcVp2 (Ag + Aj) . 



Assuming equal contributions to the lift from the angle of attack and camber 

 terms (\-A^) the total thrust on the propeller is given by 



T = 477pc^Vp2 AqNR sin a^ , 



where c^ is the effective half-chord length at 0. 7R. The thrust coefficient c^ is 

 then found in terms of the advance ratio J, defined by J = u/nD = u/2nR, with n 

 the number of shaft revolutions per second, where 



P\]^7tR^/2 



A graph of C-j. vs J is given in Fig. 39, where the value of nAq is chosen to 

 correspond to the particular operating condition Cj- = 1.0 for J = 0.7, for the 

 value Cp/R = 0.1875. These conditions are selected as an appropriate range for 

 calculations that would illustrate the nature of the results of the theoretical 

 study. 



The variation of the forces, as functions of the parameters characterizing 

 this physical problem, is determined from numerical computations for the fol- 

 lowing set of conditions: 



propeller diameter, 2R = 16 ft, 



propeller chord at 0.7 radius, 2Cg - 3 ft, 



214 



