Theory of Unsteady Propeller Forces 



The frequency of bearing forces is represented by mNQ/277 = mnj.N. Therefore, 

 denoting the components of forces and moments for frequency mnj.N > o by Kp^^^, 

 Kpym' etc., we can get , ,., 



Kr..= K^r^-^"^^^-4"r^e^"^^% Kp^,= 4;;'^e-^-^%K^7^>e^"^^^ (130) 



From Eqs. (127), (128), and (129) the complex functions Kp7^\ '^y^^\ etc., 

 are the conjugate complex functions of Kp™ , Kpy , etc., respectively. Conse- 

 quently, for mPrN > 0, denoting the amplitudes of Kp^^, Kp^^, etc., by Apxm Ap^^ 

 etc., respectively, they are equal to twice the absolute values of K^^ \ K^^'', 

 etc., respectively. That is, we get for m > 1 ■, „. 



(mN)| . _ I (mN)| _ , (mN), 



'Fx I' -^Fym - "^ I '^Fy I' ^¥ zm ' ^\^¥ z I 



V _ olvC^N)! . _ ^|„(mN)| . _ I (mN)| 



^Mxm ^ ^I'^Mx I' ■^Mym - "^ I '^My I' "Mzm^ -I'^Mz I 



(131) 



With the particular values of a propeller 0^(f), ^(^), x* (^,1^), and e, the 

 inflow velocity components v*, v* , and v*, and the advance coefficient v^ given, 

 let us calculate the bearing forces by using the preceding equations. That is, 

 calculating v (f), v*^(^), v*^(^), and0by using Eqs. (112) and (113), we obtain 

 G„(^) by solving Eq. (119) and then calculate V^^ and V^,^ from Eqs. (117), V*^, 

 V*^, V*^, V*^, and Wi(^,- S^) from Eqs. (121), and C^lC^-^) f^o^ the first of 

 Eqs. (122). On the other hand, when the values of Cpi3o(<f), ap(^), and bp(cf) are 

 assumed to be known for each blade section, Cp^ can be obtained from the second 

 of Eqs. (122). Substituting the resulting values of G^(<f), V^^, V^^, v*^, V*^, and 

 Cpo into Eqs. (123) and (124) we can calculate Kp^, Kp^, Kp^, K^,^, K^^, and K^^. 

 Thus the bearing forces can be obtained. The detailed procedure to calculate 

 Kp^ and K,^,^, i.e., C^ and Cq was presented in the appendix of Ref. 6. The com- 

 ponents of force and moment acting on the propeller other than Kp^ and K^^ can 

 be calculated according to a similar method as for Kp^ and K,^^. Similarly, by 

 substituting G„(^), V^^, V,„, V*„, and V*^ into Eqs. (128), (129), and (130), and 

 further assuming Cp^ to be constant, we can calculate the amplitudes Ap^^, Ap^^^, 

 etc., of vibratory forces and moments for frequency mnj.N (>0). 



NUMERICAL EXAMPLES OF BEARING FORCES 



As numerical examples, propeller M of Ref. 6 is adopted as a parent pro- 

 peller, whose principal parameters N, ^^,Q){^), e, etc., are shown in Table 1. 

 Then the variations from propeller M are as follows: ^m(^) ^^^ ^(^) °^ ^^Y 

 given propeller are respectively a and /3 times as much as those of propeller M. 

 The values of the particular magnification factors a and (i used here are sum- 

 marized in Table 2. For propellers Mjj, ^22* "^32' '^' '^^'^ '^52' ^^ ^^^ a = 1 

 and /3 - -0.5, 0, 0.5, 1.0, and 2 respectively, and this group is employed to deter- 

 mine the effect of skew on bearing forces. When we take a = 1 and [i - 0.75, 

 1.00, and 1.25 for propellers M^j, M, and M43 respectively and also a = and 

 (i = 0.75, 1.00, and 1.25 for propellers M21, M22, and U^^ respectively, these two 

 groups determine the effect of the blade area ratio on the bearing forces. The 

 blade area ratios of the given propellers are also shown in Table 2. 



63 



