motion. Specification of these force terms is discussed in Section 2 and Appendices D and E of this 

 report. 



The above equation is quite general since it includes both cable bending stiffness and finite ampli- 

 tude vibrations, i.e. tension fluctuations. We now will examine the relative importance of these effects 

 for marine cables. First, consider the ratio of the fluctuating tension to the equilibrium tension. 



2 To L^ " " • 



3 EA^ Y^ 



It is reasonable to assume that EA/Tq = E/a ~ 6 x 10^ and Y~ D/2 for actual cables that undergo 

 flow-induced vibrations. Furthermore L/D ~ 10^ is a conservative estimate, particularly for « > 2, 

 and this ratio therefore becomes proportional to «^ x 10~'. Using these estimates and the additional 

 fact that the moment of inertia / = (D/lV, one then finds that the ratio of bending stiffness to equili- 

 brium tension is on the order of n^ x 10~^. The justification for treating the cable as an equivalent 

 homogeneous string is thus apparent, as well as an ordering of the assumptions inherent in this approx- 

 imation. The natural frequencies for a taut cable under the homogeneous linear string assumption are 

 given by the simple classical relations 



Since the nonlinearity is small, a first approximation to the nonlinear tension fluctuation Tj can be 

 obtained by substituting the linear string equation solution into the nonlinear expression, which yields 

 the result 



If = r COS — - — sm^ (lit. (.A13J 



The transverse motion of a particular cable now can be adequately predicted if the required cable pro- 

 perties are known. The virtual mass (the sum of the structural mass and added mass contributions) 

 and damping are available, based upon recent experiments. The experimental characterization of the 

 structural damping, the added mass, and the hydrodynamic damping is discussed in Appendix C. This 

 discussion has been limited to taut cables; the dynamics of slack cables and the development of criteria 

 for delineating the two regimes are discussed in Appendix B. 



121 



