Appendix B 



DYNAMICS OF SLACK MARINE CABLES 



B.l. The Linear Theory for a Slack Cable. The vibrations of taut cables are described appropriately 

 by the classical taut string equations. This approach neglects the cable's bending stiffness and finite 

 amplitude vibration effects, but it is accurate to within 2-4 percent for many cables over a wide range of 

 conditions (see Appendix A and references Bl and B2). As the tension is relaxed, a cable eventually 

 assumes the configuration shown in Fig. 81. //is the horizontal component of tension at the supports 

 and each vertical component V is equal to half of the total cable weight. The limiting sag-to-span ratio 

 s//— ► is accompanied by // = 7 since the cable weight becomes a negligible fraction of the tension. 

 At the other extreme, when si I becomes large, V is comparable to or larger than H and the cable 



assumes a classical catenary shape. The natural vibrations of catenaries are known (83) for — > 1:10, 



but until recently they could not be reconciled with the taut string theory as the ratio of sag to span 

 vanished. This difficulty has been overcome by Irvine and Caughey (84) as a result of including the 

 extensional behavior of the cable in the theory. 



A summary is given here of Irvine and Caughey's recent development and the results applicable 

 to marine cables are discussed. The equilibrium shape of an inextensible cable is given by 



2^ 



^ 2H 



(Bl) 



1 + 4 



d 



2 



3 



I 





^ / 



for d/l < 1:8 where d = mgl^/^H is the midspan sag. The length of this cable is L = / 



, so that if three of the quantities mg, //, d, I and L are known, the other two can be 



found. However, owing to stretch, the sag and length of a real cable are greater than the inextensible 

 values while the horizontal component of tension of the stretched cable is less. If this new sag is s 

 while the new horizontal component of tension is (H — A//), then equilibrium dictates that 



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