Milgram 



This can be written in terms of the angular variable by use of Eqs. (8) and (10); 



^7T CO Kb °° 



L = PUq b^ Z_j ^n '^O ^i" ""^ sin0 - sin 20 2_, ^n '^ ^ " ""^ '^'^ • ^^^^ 



•^0 n= 1 " "* n= 1 " 



Carrying out the indicated integration gives 



vu \ 



(32) 

 The heeling moment about the midspan is called M , and is given by 



Kb 

 L = /OUo b' — lU. A, A 



.b/ 2 

 M. = I e (y) y dy . 



■b/ 2 





(33) 



Using Eqs. (7), (8), and (11), 



pUo b-^ 



CO ^7T 



n= 1 "^0 



Kb 



Uq sin niA sin 20 - (cos0 - cos 30) sin n0 



4 



d0 , (34) 



which reduces to 



M = 



A, U„ 77 - 2Kb 



L ^2n 



n=l \4n -1 4n -9, 



The heeling moment about the base of lifting line is 



(35) 



M^ = M + L 



(36) 



For the relative values of Kb and Uq commonly encountered, the dominant 

 terms contributing to the heeling moment are the lift itself which are found in 

 the second term in Eq. (36) and the first term of Eq. (35). 



Conclusions from the Theory of a Single Lifting Line 



The second term of Eq. (32) represents the effect of the nonxmiform 

 strength of the incident wind on the lift. For the case of a constant windspeed 

 gradient considered here, the lift is affected through the second term in the 

 Fourier series representation for the circulation distribution. Different forms 

 of nonunlformity would affect the lift through other terms in the series. It 

 should be noted that the ratio of the part of the lift generated by the nonuniform 

 part of the wind to the lift generated by the uniform part of the wind is small. 

 This ratio can be obtained from Eq. (32) as 



1408 



