Here F and G ceui be expressed in terms of real amplitudes A and B as: 



F = Ae'^^S G = Be''^^ [29e,f] 



Furthermore, Equations [27] and [28] can also be written more 



iojt 

 explicitly because of the assumed time variation as e , thus: 



2- 2- 



a V EI 3 M 

 M = EI (1 + iwn) — 2 '*' KAG — 2 ^^'^°^^ 



9x 9X 



2— 

 (- yto + iucy) v + — - = [28a] 



8x 



An equation for M alone can also be obtained by differentiating 



Equation [28a] relative to x and substituting from the resulting equation 



2- 2 

 for 3 v/3x in Equation [2Ta]. Slightly rearranged, the result is: 



k- 2- 



EI 1 + icon 9 M ^ E^ 3fM _ ^ ^ q 



2 , w^/ N , 1+ KAG ^ 2 

 yo) 1 - ilc/toj 3x 8x 



Now it is convenient to introduce the notation: 



^ ycij^ , EI 2 



q = - — ; C = 1 



EI 2KAG 



1 - i(c/ai) . _ 1 - en . (c/oi) + trin 



1 + iu)n = e - ig - 2 2 - \ ^ 2 2 



i + wn i + wn 



The symbols e and g represent real quantities whose values can be read 



from the last member of the equation, and always g > 0. The equation ob- 



- h , s 



tained for M, multiplied through by q (e - ig) , then reads: 



^ + 2cq^ (e - ig) 1^ - q^ (e - ig) M = [30] 



3x^ 3x^ 



30 



