A = 



Ship Maneuvering in Deep and Confined Waters 



'cosijjcosB -sin4iCos<(> +COS ijjsin 9 sin<|> sinilj sin<() +cos i|j sln9 cos4> 

 sin^jcosB cos 4j cos<}> +sini|jsin9 sin<{> -cos 4j sLn^ +sin4j sin 9 cos <}) 

 -sin 9 cos 9 sin 4) cos 9 cos (f) 



(4.2) 



When applied in opposite direction the transformation is 



Xp==A (x^p- xj =A(x^p- x^ 



(4.3) 



where A is the transposed matrix, in which rows and columns 

 appear in interchanged positions. 



In particular, note that the gravity vector g^ = gz^ will be 

 given by the column vector 



g = A 



- g • sin 9 

 g cos 9 sin ^ 

 g cos 9 cos 1^ 



(4.4) 



in the moving system. 



From Fig. 7 will be se^n how the absolute (total) value of the 

 time derivative of any vector h In the body system may be calcu- 

 lated from the relation 



h^j,, = h+0 xh 



(4.5) 



The angular velocity vector J2 may now be expressed in 

 terms o_^the Euleriaij angles and their time derivatives: For the 



vector h there is h^ = Ah and 



h"^ = A(A"h + aT) sh" + AAh" 

 a OS 



(4.6) 



and so the column vector O is obtained from the CQjj-esponding anti- 

 symmetric angular velocity matrix for the product 



Qjrei 

 AA, 



825 



