

vector of the second derivatives. 



angular velocity vector expressed in coordinates 

 along the body axes. 



In order to express the angular position of the structure 

 with respect to the fixed inertial coordinate system, it is neces- 

 sary to utilize a set of three Euler angles. These are defined as 

 follows; let the body axes initially coincide with the axis system 

 fixed in space. Figure 1. The structure first rotates in yaw 

 through angle 6_ about oy , then through the pitch angle, 6-,, about 

 the new position of oz, and finally, through the roll angle 6, 

 about the final position of ox. The relationship between the two 

 coordinate systems is now: 



\ Y 



cos9„cos6-. 



-cosG cos6«sin6 



sino^ coso„sint 

 +cos6, sin8„ 



CO so, CO so. 



-sinO-cosD^ 



cosQ sin6 sin0„ 



+sinG,. cos6„ 



cos9 cos6„ 



-sin9, sin0„sine, 

 1 Z 



y-x2y (8) 



Now, if {co} = (o 



'1' ""2' 



ud^) are the components of the instantaneous 



angular velocity along the body axes, the relationship betv/een {oj} 

 and the time derivatives of the Euler angles is 



{oj} = [B]{0} 



where 



[B] 



sin6 , 







cosO, cos6 T 



sinS -, 



-sin6-, cosG -. 



cos6 I 



(9) 



In the case of small angular motions, we see that sin6 . ~ 6. (small) 

 and cos6 . = 1 , so 



[B] 



(10) 



135 



