Hanging cables with small bending stiffness

Abstract

IN THIS paper we study the problem of the equilibrium configuration of a cable suspended between two fixed supports. The only load on the cable is its own weight. We model the cable as a Cosserut rod, and assume the cable may undergo extension and flexure but not shear. We wish to study cables having small bending stiffness, which we do by introducing a small parameter E into the constituative equations (stress-strain laws) for the cable. When E = 0 the equilibrium equations reduce to those which arise when the cable is modeled as a string with no resistance to bending. These equations have been analyzed by Antman [l]. He shows that there is always exactly one solution of this problem in which the cable is in tension. The problem for small positive E is a singular perturbation of the string problem. For our problem, based on the rod model, in addition to stipulating the position of the ends of the cable, we must impose additional boundary conditions. We impose the condition that the cable is clamped at both ends. The solution to the string model will not satisfy these additional conditions. Thus we would expect that as E tends to zero the solution of the rod model will tend to the solution of the string model in the interior of the cable. However, in order to satisfy the additional boundary conditions boundary layer corrections must be included to obtain an approximate solution to the rod problem for small E. In a previous paper [2] we considered a similar problem, that of a conducting wire in a magnetic field. This problem was chosen as a starting point for a theory of rods with small bending stiffness because the corresponding reduced (string) problem can be solved explicitly and the solution has a simple form. The main point of that paper was to show how to compute the boundary layer corrections starting from an ad hoc ansatz. There was no claim that the solution constructed there was actually asymptotic to an exact solution as E tends to zero. After completing this paper we became aware of the work of Schmeiser [3] and Schmeiser and Weiss [4]. These papers (and the references cited therein) deal with a general theory of singular perturbations for systems of nonlinear ordinary differential equations. In these papers it is proven that, in fact, the approximate solutions constructed are asymptotic to an exact solution. It became clear that our work could easily be put into the framework of this theory. We also realized that our technique was far more general than originally thought and could be applied to problems more complicated than our model problem addressed in [2]. In Section 2 we will describe the rod model and our concept of small bending stiffness. In Section 3 we will describe the string model and its solution. In Section 4 we will state the general theorem on singular perturbations for nonlinear systems of ordinary differential equations and in Section 5 we will apply this theorem to our problem. Section 6 consists of some concluding remarks.