Weighted catenary

A weighted catenary is a catenary curve, but of a special form. A "regular" catenary has the equation

A hanging chain is a regular catenary — and is not weighted.

for a given value of a. A weighted catenary has the equation

and now two constants enter: a and b.

Significance

A catenary arch has a uniform thickness. However, if

  1. the arch is not of uniform thickness,[1]
  2. the arch supports more than its own weight,[2]
  3. or if gravity varies,[3]

it becomes more complex. A weighted catenary is needed.

The aspect ratio of a weighted catenary (or other curve) describes a rectangular frame containing the selected fragment of the curve theoretically continuing to the infinity. [4][5]

The St. Louis arch: thick at the bottom, thin at the top.

Examples

The Gateway Arch in the American city of St. Louis (Missouri) is the most famous example of a weighted catenary.

Simple suspension bridges use weighted catenaries.[5]

References
  1. Robert Osserman (February 2010). "Mathematics of the Gateway Arch". Notices of the AMS. Missing or empty |url= (help)
  2. Re-review: Catenary and Parabola: Re-review: Catenary and Parabola, accessdate: April 13, 2017
  3. MathOverflow: classical mechanics - Catenary curve under non-uniform gravitational field - MathOverflow, accessdate: April 13, 2017
  4. Definition from WhatIs.com: What is aspect ratio? - Definition from WhatIs.com, accessdate: April 13, 2017
  5. Robert Osserman (2010). "How the Gateway Arch Got its Shape" (PDF). Nexus Network Journal. Retrieved 13 April 2017.

On the Gateway arch

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