Optimization of the catenary and its modifications

The paper deals with the optimization of an ordinary inextensible catenary in a homogeneous gravitational field near the Earth’s surface and its modifications, viz. a stretchable catenary and a catenary in the Newtonian field. The optimization criterion is tension force minimization at the suspension points of the catenary where it reaches its highest value among all the catenary points. This is ensured by choosing the catenary optimal length. For all considered versions, analytic solution of the optimization problem based on the expression for the catenary tension at the specified points is constructed. In the course of a solution, it is possible to obtain fairly simple final expressions and give visual graphic illustrations. Besides, it is shown that for catenary modifications in limiting cases, we obtain solutions that are completely consistent with the version of the ordinary catenary. The obtained results are of theoretical interest and of great practical importance for designers and builders of power lines, cable cars in high altitude areas, various long length cable structures and other chain systems.