Propeller-Induced Appendage Forces 



where u(x, t) is obtained as the real part of Eq. (4). The forces on the appendage 

 due to this dipole distribution are determined from the unsteady Lagally theorem, 

 which includes a quasisteady term and a nonsteady term. The resulting forces 

 are given by 



dZ2 



dx 



Z= X 



' , . X3 - iYj = - l-np /i.(x, t) 



■« 



for the quasisteady term and 



X^ + iY, = -277P A^(x,t) dx : ■:■'■•.;..•., , (5) 



-f-l, B t .-■■:-; 



for the unsteady term, which is found in this case to be only an axial force on the 

 appendage. - - -:;? ,r, .; - 



Examination of the expression for the quasisteady dipole term shows that 

 the force magnitudes are expected to be small and to produce higher harmonics, 

 since the dipole strength is proportional to the induced complex velocity and it 

 is multiplied by the velocity gradient. Thus this term will be deleted from fur- 

 ther consideration, since numerical evaluation has also shown that it provides a 

 negligible contribution. The unsteady axial force term arising from the dipole, 

 given by Eq. (5), can be expressed as 





n=l 



.N-^c/^e/Nm^ . {\ ^ g-Nmj/R,/Nm_P ^ ^ 



from which it is possible to separate the effects of propeller vorticity and thick- 

 ness as in the previously derived force expressions. 



Since the appendage is also represented by an unsteady vorticity distribution, 

 a leading-edge axial suction force will arise. This term will also be small, since 

 it is proportional to the square of the induced vorticity, and hence the induced 

 velocity field, as well as contributing only higher harmonics. Therefore it will 

 be neglected when considering numerical evaluations. 



To apply the previous results to an actual three-dimensional propeller- 

 appendage combination, it is necessary to include two infinite cascades of blades. 

 Thus, as the blades on one side of the vertical plane through the propeller axis 

 are moving down past the appendage, the blades on the other side of the plane are 

 moving up. The previous expressions will hold if the directions of certain veloc- 

 ities, angles, etc., are reversed; and an analysis was made in terms of the ef- 

 fects of the different velocities when separately considering odd-bladed and 

 even-bladed propellers. The details of the required analysis are presented in 

 Ref. 5, and the conclusions obtained by considering the effects of the two cas- 

 cades are summarized as follows: 



213 



