yave Induced Forces and Motions of Tubular Structures 



rotation may be expressed by a-^- such that 



Uj = QTij Uj 



Uj =Q'ijUj . (12) 



The components of force on an element of length, dx, In the oxyz- 

 dlrections are given by 



dF, = (iljj"ij + Xj.u.) cbE, j= 1,2,3 (13) 



where \i-. and X.; are added mass and linear drag coefficients. For 

 the cylindrical member, only ^22* ^33' ^22' ^^^ ^33 ^^® noja- 

 zero, while for the point volume all of the diagonal terms in |x and 

 X. are nonzero. 



The forces may now be expressed in o^n^ t>y the inverse of 

 the above transformation 



dF ='a-. dF. 

 I ji J 



= (iXj.u. + \..xi.) dx . (14) 



Here, the added mas^s_and drag coefficients In oir\l, are seen to be 

 related to those In oxy z by the following expressions. 



^\\ "jinj ,j 



\.. = a.,\..a.. . (15) 



Corresponding expressions for the elementary moments 

 about oxyz and o|il^ maybe written. The total forces and 

 moment may be obtained by Integrating these expressions over the 

 length of the member, noting that Uj and Uj contain terms with 

 the same trigonometric functions of time which appear In the Froude- 

 Krylov buoyancy Integrals, (11), and may. In fact, be combined with 

 them In carrying out the Integration over the nnember length. 



Evaluation of these integrals yields a set of forces and 

 moments In o^ti^, A coordinate translation and rotation may now be 

 applied to express these forces and moments In the space coordinate 

 system oxyz. 



1093 



