A catenary-like cable confined in a circular cylinder
DOI:
https://doi.org/10.31349/RevMexFisE.21.010211Keywords:
Catenary; flexible cable; variational calculusAbstract
The problem of obtaining the curve of a cable suspended between two points, supporting only its own weight, was solved simultaneously, in the 17th century, by Johann Bernoulli, Leibniz, and Huygens. This curve is called the catenary. This article solves a modified problem in which the suspended cable is confined in a vertical cylinder. For this, an functional is formulated to describe the potential energy of a fixed-length confined cable in any possible arrangement. Then, the variational problem of extremizing this functional is presented and the Euler-Lagrange differential equation is deduced. The analytical solution of this equation is obtained for the cable suspended by two points at equal and different heights. Furthermore, the tensile force acting on the cable is determined. Numerical results are presented comparing the effect of confinement on the tensile force in relation to the traditional catenary.
References
James L. Meriam and L. G. Kraige. Engineering Mechanics: Statics. Wiley, 7th edition, 2011.
Sirlene Rezende Faria. A catenária. Universidade Federal de Minas Gerais, 2011.
Eric W. Weisstein. Catenary. from mathworld - a wolfram web resource. http://mathworld.
wolfram.com/Catenary.html. [Acessado em 5 de Fevereiro de 2020].
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Mexicana de Física E 17(2), 276-284, 2020.
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Variations. Mir Publishers, 1975.
It is known in mechanics that by minimizing the potential energy of the cable, we are finding
the shape of the cable in static equilibrium.
There is another approach to finding the static equilibrium configuration of the cable. Determine
the coordinates of the center of mass ( ̄X, ̄Y , ̄Z) of the cable. Then minimize the coordinate ̄Z,
because in, a situation of static equilibrium, greater stability means less value for ̄Z.
Steven C. Chapra and Raymond P. Canale. Numerical Methods for Engineers. McGraw-Hill,
th edition, 2014.
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