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CATENARY OF A SURFACE

Notion studied by Bobillier in 1829.
See: Paul Appell: cours de mecanique rationnelle, page 220. |

Differential equation: , where is the normal vector of the surface. |

The catenaries of a surface are the equilibrium lines of an inelastic flexible homogeneous infinitely thin massive wire included in the surface, placed in a uniform gravitational field.

The catenaries are also the curves that minimize the altitude of their center of gravity, which means that the catenary with length *l* that joins *A* and *B* is, among all the homogeneous curves with length *l* joining *A* and *B* traced on the surface, the one with center of gravity at the lowest altitude.

Here is the derivation of the above differential equation:

We write the force equilibrium relation for a small element of the wire: (
= tension of the wire at M, = weight,
= normal reaction of the surface), which gives: (1)
(m = linear mass density of the wire,
=
vector tangent to the curve). But applying the Binet formulas, we get: where is the normal vector of the surface. Multiplying by yields , i.e. , hence ; by substituting this value for T in (1), we derive the differential equation. |

Examples:

- the catenaries of a non horizontal plane are the usual catenaries.

- the catenaries of any vertical cylinder are the curves that develop into catenaries with vertical axis.

- the cylindrical catenaries

- the conical catenaries

- the spherical catenaries

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© Robert FERRÉOL 2018