\lave Induced Forces and Motions of Tubular Structures 



in terms of time dependent deviations from this mean position. The 

 linear displacements of the center of gravity of the platform from its 

 mean position may then be expressed by the small quantities x(t) , y(t), 

 z(t), measured in the oxyz system. We next express the rotational 

 motion of the platform in terms of the Eulerian angles ci{t) , |3(t) , y{X) . 

 These angles are so defined that the angular displacement between the 

 two coordinate systems, oxyz and OXYZ, may be created by imagin- 

 ing the platform as first oriented such that the two coordinate systems 

 coincide. It is then rotated about OX through the angle a, then about 

 the new position of OY through the angle (3» and finally about the new 

 position of OZ through the angle y to bring the platform to the final 

 position of angular displacement. For small values of a , (3, y, the 

 two coordinate systenns will now be related by: 



1 



(1) 



The equations of motion may now be written. It is convenient to 

 first write the equation for translatory motion in oxyz, thus it is 

 assumed that all forces acting on the body have been expressed* 

 in this system, giving 



f, = mx. , 



i= 1, 2, 3, 



(2) 



where the Xj = x(t) , y(t) , and z(t), respectively. 



The equations of angular motion may be most easily written 

 in the body coordinates, OXYZ, since the moments and products 

 of inertia of the structure are constant in this systenn.. If the exter- 

 nal moments are also expressed in OXYZ, the Euler equations for 

 rotational motion in rotating coordinates are obtained: 



M: 



= ^ Ijjttj +0(^2(0^), i= 1, 2, 3. 



(3) 



j=i 



Here, 



0. = components of the angular velocity vector in OXYZ, 



I.. = moment of inertia about i-axls If 1 = j, 

 ij >' ' 



i.. = (-) product of Inertia If 1 =5^ j. 



Note that the transformation expressed by Eq, (1) can be applied to 

 forces and velocities as well as to coordinates. 



1087 



