Three-Dimensional Boundry Value Problems for Flexible Cables
Authors: Bedendender John W. 
Published: October 2008

ABSTRACT
In the design of systems towed or moored in a fluid by a flexible cable, boundary conditions are often specified at two different points along the cable. These boundary value problems are commonly solved for the equilibrium position of the cable by integrating a family of initial value problems and then selecting the desired solution.
An iterative solution technique is presented by which boundary value problems in two and three dimensions can be solved directly. The differential equations of cable equilibrium are derived for a three-dimensional velocity field. The hydrodynamic loading functions are expressed in general forms so that the most appropriate hydrodynamic force model for a given cable application can be used, or solutions for different models can readily be compared.
The differential equations are solved by a digital computer program which incorporates the solution technique for boundary value problems. The force components at the cable terminations necessary to achieve the given boundary conditions are obtained from the solution; for example, the hydrodynamic forces which a towed body must exert on its towing cable to maintain a specified position can be determined. Examples of solutions to two and three-dimensional problems with space variable velocity components are given.

INTRODUCTION
Flexible mooring and towing cables have had wide application as components of hydrodynamic and aerodynamic systems. Clearly, an understanding of the behavior of these cables is imperative for intelligent design of the complete system. The objective of this paper is to present different solution techniques which will provide a more direct solution to certain flexible cable problem s than was previously possible.
The two-dimensional cable problem has been studied by a number of investigators. The most significant difference in the various investigations is the mathematical model used for the hydrodynamic forces acting on the cable, i.e., the hydrodynamic loading functions.
Pode 1 provided a comprehensive report on the subject, and his solutions are still commonly used. He assumed a sine-squared relation for the normal component of hydrodynamic force and a constant for the tangential force component. The solutions were put in the form of integral "cable functions" which were evaluated numerically and tabulated. Knowing the tension and slope at one point on the cable, the conditions at other points along the cable can be found using Pode' s functions.
Whicker2 added to Pode' s work by developing cable functions using a modified hydrodynamic force model applicable to faired as well as round cables.
The most complete reference to date on the solution of tow cable problems by the use of cable functions has been published by Eames3. He provides an extensive discussion on the form of the hydrodynamic loading functions as related to the physical conditions of the problem. Cable s are classified according to the relative significance of the various terms of his proposed loading models. This reference is particularly valuable for aiding in the selection of a hydrodynamic force model.