Paulling 



Now, Mj may be transformed by (1) into components, nrij, 

 expressed in the space coordinate system. We note, further, that if 

 the angular velocities , O; , are small quantities , the moments , rrij , 

 causing the motions must oe small of the same order. The transfor- 

 mation of moments, therefore, after dropping products of small 

 quantities , yields 



mi=Mi. (4) 



Similarly, the coinponents of angular velocity, $2:, in OXYZ 

 may be transformed into the space coordinate system. For small 

 angular velocities , this results, approximately. In 



'a. = dj . (5) 



Here, the a-^ are the components in oxyz of a small rotation 

 of the structure about an instantaneously fixed axis in space. In 

 other words , for the small angular motions to which we limit the 

 present analysis, the Euler angles are approximately equal to the 

 components of the body rotation during a small time interval about 

 the fixed space axes. 



We may now write the translational and rotational equations 

 of motion. In the fixed coordinate system. In the combined form 



'j = ) ^ij'^j I = 1 , 2, . , . , 6. (6) 



Here f . , x.; I = 1, 2, 3, are the forces and displacements In the 

 X, y, z-dlrectlons , f. , x.; I = 4, 5, 6, are the moments and rota- 

 tions about the x, y, z-axes, 



m.. , m„„, m,, = m = mass of structure, 

 II 22 33 



m , m , m = moments of Inertia about XYZ-axes, 

 44 55 66 



respectively. 



-m 



'"«='"5.= -\\ \ ""^-^ 





YZ dm Products of Inertia about 



the OXYZ -axes. 



m^g = mg^ = - \ \ \ XZ dm 



1088 



