Catenary Curve Calculator

Catenary Definition

A cable takes on a catenary shape when it is supported at both ends and is only pulled in one direction by its weight. As a result, the cable’s center dips due to the uniform gravitational force, creating a symmetrical curve on both sides of the minimum point. The catenary curve is intriguing because it frequently appears in the environment.

A natural catenary has two functional characteristics:

1. The horizontal force (Fx) in the cable is constant along its entire length; and
2. The vertical force (Fy) in the cable is equal to the weight of the cable being carried at any given position (i.e. Fy is zero at the bottom of the loop).

Resolving these forces yields the angle of the wire at any point, for example,

\( tan^{-1}(\dfrac{Fy}{Fx}) \)

Equation of the catenary curve:

\( cosh \space x = \dfrac{e^x + e^{-x}}{2} \) OR \( y = a \space cosh(\dfrac{x}{a}) + b \)

here;
x is the abscissa and y is the ordinate of a point on the curve
e is Euler’s number,
a is a constant: horizontal tension divided by the weight per unit length
And b is the integration constant impacting the y-intercept of a graphed catenary curve.

Applications of the catenary curve

Catenaries are used in engineering and architecture, for instance, to create hanging bridges or arches when they are turned upside down. The St. Louis Gateway Arch is one of the most striking examples. Catenaries are also present in the natural world, such as the curvature of a spider web.

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