Propeller -Induced Appendage Forces 



where f (Xp) is the propeller profile function, chosen for the present purposes 

 as a symmetric blade with a parabolic profile given by 



fC^o) 



Following the same procedure as for the vorticity distribution, the complex 

 velocity induced at the appendage by the source distribution of the propeller 

 blades can be found, and the total nonsteady complex velocity induced at the ap- 

 pendage by the propeller blades, including both the vorticity and the thickness 

 effects, can be expressed as 



dW 

 dZ 



L'K 



mNx/R. i mv„ t 



(4) 



where 



G^ + V 



m 



27tE "1 P 77N 



G * . 



with G^ defined in Eq. (3) and 



t _ 1 -i2ap-cr7Tm[(Xo/c)-sin a ^] 





■anme " (i + a77me'°P] 



Using the expressions for the complex velocity induced at the appendage, the 

 normal velocity is given by 



'(x,t) = -Im ^ 0„ 



mNx/R imi'^ t 



SO that it is expressed as an infinite sum of components, each of which is of the 

 form 



V = e 



- iu„x/U imv„ t 



where 



f^rr. = 



JmNU 

 R. 



Using the results of unsteady airfoil theory (13), the nonsteady transverse force 

 (lift) on the appendage due to the propeller blade vorticity and thickness is rep- 

 resented by a sum of two terms: Yj" due to the propeller blade vorticity and Y/ 

 due to propeller blade thickness. These terms are given by 



211 



