to first order, where Pq is the first term of Equation [ll]. 



The moments are obtained by calculating the force per unit length 

 along each spar, multiplying by the appropriate lever arm, and integrating 

 along the lengths of the spars. It should be noted again that the monnents 

 are calculated with respect to the space-fixed axes. Thus a point located 

 at z^' on the i -axis has space-fixed coordinates (see Equation [!'])• 



X = cLr, cos 6- + Xp, + (3 z.' - y an sin 9. ; 

 U 1 "^ 1 ' 1 ' 



y = aQ sin 9^ + yQ + y aQ cos 9- — a z^ ; 

 z - z^ + Zq + a a.Q sin 9- — (3 a^ cos 9- . 



FIRST -ORDER EQUATIONS OF MOTION 



Let M be the mass of a spar. (The N spars are assumed to be iden- 

 tical. ) Let I^, J^, Kj be the moments of inertia of the i spar about the 

 X-, y-, z-axes, respectively. Moreover, let Mq, Iq, Jq, and Kq denote 

 the mass and moments of inertia for any additional superstructure, and 

 assume (0, 9, Zq) to be the center of gravity thereof. Then the equations 

 of motion are 



N 



(Mq + NM)iQ = ^^X. [15a] 



i = l 



N 

 (Mq + NM)yQ = ^^i [15b] 



1=1 



N 

 (Mq + NM)zq = ^ Z. - g(NM + Mq) [15c] 



i=l 



N N N 



- MogZQ+ a^^li = ^^A. - g ^^ J m(z.')ydz.' [I5d] 



i=9 i=l 1=1 ^ 



15 



