Appendix A 

 DYNAMICS OF TAUT MARINE CABLES 



A.l Equation of Motion for a Taut (Stretched) Cable. Consider a uniform cable stretched between 

 rigid supports a distance L apart with an equilibrium position along the x axis and equilibrium tension 

 Jo- The cable has a virtual mass density p, cross-sectional area A, elastic modulus E, and a moment of 

 inertia / about the neutral axis z. For simplicity it is assumed transverse oscillations take place only in 

 the xy plane. To account for the damping, we assume a term in the equation linearly proportional to 

 the transverse velocity by a damping coefficient C,. The coefficient ^ is taken as the damping of the sys- 

 tem as measured in still air. This is the usual approach to specifying the structural damping. A further 

 discussion of structural and hydrodynamic damping is given in Appendices C and E. Longitudinal dis- 

 placements are neglected and the transverse displacefnent at the position x is taken to be y{x,t). 

 Further, for typical cables and the frequencies of flow-induced vibrations, shear deformations and rotary 

 inertia are negligible. The potential energy of the cable is then a sum of bending and stretching poten- 

 tial energies. From elementary beam theory the bending energy is given by 



xnCEI^dx. (Al) 



Jo Qx^ 



The approach developed by Murthy and Ramakrishna (Al) is employed here to determine the 

 stretching contribution for the nonlinear vibrations of strings. This is a good approximation for cables 

 that undergo flow-induced "strumming" motions in water. As an element of cable dx is deformed into 

 the planar element ds, the stretched length of the element is 



1/2 



^5 = {dx^ + dy'^y'^= dx 

 and the local strain is given by 



1 + 



ihl) 



ds — dx 



1 + 



19 



1/2 



- 1. (A3) 



