Theory of Unsteady Propeller Forces 



versus angle 0. The symbol i as expressed by <f = .f^ (i = 1, 2, . . . , 7) in the figures 

 and tables of this section designates the blade section, and the relation between i 

 and ^i is defined in Table 1. Denoting the advance coefficient by standard nota- 

 tion and the circulation about a blade section by J and G^(^", s) respectively we 

 get 



J=rrv,, G^(,^s) = G,(^, -Sk) = T G„(^)e'"'''. (133) 



Further we adopt 0.276 as a convenient approximate value of v in the numerical 

 calculation. Following the procedure in the previous section the values of G^(^) 

 for various values of J , and also v^^, v^^, v*„, v*^, v*^, v*^, WJ(^^ - s^), 

 G^(^,s) , C^L(<f,s), etc., can be calculated. Then, denoting the zero lift angle of 

 the blade section in radians by agi and using the relations 



CpDoC'^) = 0.0080 , ap(.f) = 0.08315 , bp(<f) = 6.283 a^^ - 0.1097 (134) 



in accordance with Kerwin (6), the values of Cpp and consequently the bearing 

 forces Kp^, Kpy, Kp^, K^,^, K^jy, and K^,^ can be calculated numerically. Then, 

 denoting the mean values of Kp^, Kp^, etc., with respect to time s by Kp^, Kp^, 

 etc., respectively and denoting the magnitudes of the vibrating parts of Kp^, Kpy, 

 etc., with time s by AKp^, AKp^, etc., respectively, the mean values can be ob- 

 tained by . 



27T/N 27T/N '.:■ - 



Kfx = ^ J Kp, ds , Kpy = 1^ j Kpy ds , : - etc. . 



-'o -'o ■■'■■ ■- . 



AKp^ = max Kp^^ - min Kp^^ , AKpy - max Kpy - min Kpy , etc. (135) 



Further, assuming that Cp^ is a constant and equal to 0.01 instead of using 

 Eqs. (134), we can calculate the aplitudes Ap^^, Apy^, etc., for m = 1, 2, 3, 4. 



We will apply the method of comparing the bearing forces of propellers on 

 the basis of the idea of thrust identity. With the propeller diameter D, the ship 

 speed V, and the mean thrust T given in common, we will compare the number 

 of revolutions of the propellers per unit time and the other characteristics of the 

 bearing forces. We will take two propellers Mj and M2 and will use the sub- 

 scripts 1 and 2 referring to propellers Mj and Mr, respectively. Then, from the 

 requirement of constant thrust, we obtain 



f = -pD'v'r.'K^^^ai) = -pD'v2j;2Kp^^(j^) , (136) 



where 



Ji = V/(n^iD) , J^ - V/{n^p) - J,/^, M = n^^/^rl ■ ^^^T) 



and Kp^i(Ji) and Kp^2(J2) means that K^^j and K^^^ are respectively functions 

 of J J and jj. Hence, from Eqs. (137) we get 



65 



